This document provides instructions and objectives for a physics laboratory practice on rotational dynamics. It defines key concepts like torque, moment of inertia, and conditions for rotational equilibrium. It also lists the required equipment and materials, such as a vernier caliper, flexometer, masses, springs, and tables. The activities to be performed involve measuring variables like time and distance to calculate acceleration and friction forces for different materials. Tables will be used to record the measured data and calculated values.
This document provides instructional materials on analyzing motion using vectors. It begins with definitions of motion and discusses rectilinear, parabolic, and circular motion. Rectilinear motion is analyzed using position, velocity, and acceleration vectors. Parabolic motion results from horizontal rectilinear motion combined with vertically accelerated motion. Circular motion is described using angular position, velocity, and acceleration. Examples are provided to demonstrate analyzing different types of motions using vectors.
The document discusses simple harmonic motion (SHM) and presents the equation of motion for an object undergoing SHM. It defines key terms like equilibrium position, spring constant, angular frequency, amplitude, phase, period, and frequency. Examples are given of finding the equation of motion for objects attached to springs given their mass, spring constant, initial position and velocity. The equation of motion is derived from Newton's second law as a second order differential equation. Its general solution is presented in the form of trigonometric functions with constants determined by the initial conditions.
How to Prepare Rotational Motion (Physics) for JEE MainEdnexa
ย
The document discusses the cross product, torque, rotational motion, and angular momentum. It defines the cross product of two vectors A and B as a vector C perpendicular to both A and B with magnitude ABsinฮธ. It describes properties of the cross product including being anti-commutative. It also defines torque as a measure of the tendency of a force to cause rotational motion, and discusses rotational dynamics and angular momentum.
Rotational motion. The motion of a rigid body which takes place in such a way that all of its particles move in circles about an axis with a common angular velocity; also, the rotation of a particle about a fixed point in space.
This document discusses key terms and equations related to rectilinear motion. Rectilinear motion refers to motion along a straight line. Kinematics deals with the motion of bodies without considering forces. Important concepts discussed include displacement, average and instantaneous velocity, acceleration, distance traveled, and equations of motion. Graphical representations of motion using velocity-time graphs are also presented for different scenarios including uniform velocity, variable velocity from rest to a final velocity, and variable velocity between two points.
1) Rotational motion and translational motion share many similarities, with position, speed, velocity, acceleration, and momentum all having analogous parameters between the two types of motion.
2) Torque is the rotational equivalent of force - torque generates angular acceleration as force generates linear acceleration. The moment of inertia of an object resists changes in angular motion, similar to how mass resists changes in linear motion.
3) For a given total mass, an object will be easier to rotate if its mass is located closer to the axis of rotation, as this lowers its moment of inertia compared to a mass distributed farther from the axis.
This document summarizes key topics in linear motion including:
1) Linear motion concepts like position, displacement, velocity, and acceleration are discussed. Different reference systems are also introduced.
2) Equations for displacement, velocity, and acceleration are provided including how to calculate these values using derivatives of position over time.
3) Specific linear motion examples like uniform and non-uniform accelerated motion are examined along with example problems.
This document discusses rectilinear motion of particles. Rectilinear motion refers to motion along a straight line, with position defined by the distance x from a fixed origin O. Velocity is defined as the rate of change of position with respect to time. Acceleration is defined as the rate of change of velocity with respect to time. Displacement refers to the change in position, while distance traveled is the total length of the path traveled over a time interval. The document provides mathematical definitions and derivations for these key concepts in rectilinear motion.
This document provides instructional materials on analyzing motion using vectors. It begins with definitions of motion and discusses rectilinear, parabolic, and circular motion. Rectilinear motion is analyzed using position, velocity, and acceleration vectors. Parabolic motion results from horizontal rectilinear motion combined with vertically accelerated motion. Circular motion is described using angular position, velocity, and acceleration. Examples are provided to demonstrate analyzing different types of motions using vectors.
The document discusses simple harmonic motion (SHM) and presents the equation of motion for an object undergoing SHM. It defines key terms like equilibrium position, spring constant, angular frequency, amplitude, phase, period, and frequency. Examples are given of finding the equation of motion for objects attached to springs given their mass, spring constant, initial position and velocity. The equation of motion is derived from Newton's second law as a second order differential equation. Its general solution is presented in the form of trigonometric functions with constants determined by the initial conditions.
How to Prepare Rotational Motion (Physics) for JEE MainEdnexa
ย
The document discusses the cross product, torque, rotational motion, and angular momentum. It defines the cross product of two vectors A and B as a vector C perpendicular to both A and B with magnitude ABsinฮธ. It describes properties of the cross product including being anti-commutative. It also defines torque as a measure of the tendency of a force to cause rotational motion, and discusses rotational dynamics and angular momentum.
Rotational motion. The motion of a rigid body which takes place in such a way that all of its particles move in circles about an axis with a common angular velocity; also, the rotation of a particle about a fixed point in space.
This document discusses key terms and equations related to rectilinear motion. Rectilinear motion refers to motion along a straight line. Kinematics deals with the motion of bodies without considering forces. Important concepts discussed include displacement, average and instantaneous velocity, acceleration, distance traveled, and equations of motion. Graphical representations of motion using velocity-time graphs are also presented for different scenarios including uniform velocity, variable velocity from rest to a final velocity, and variable velocity between two points.
1) Rotational motion and translational motion share many similarities, with position, speed, velocity, acceleration, and momentum all having analogous parameters between the two types of motion.
2) Torque is the rotational equivalent of force - torque generates angular acceleration as force generates linear acceleration. The moment of inertia of an object resists changes in angular motion, similar to how mass resists changes in linear motion.
3) For a given total mass, an object will be easier to rotate if its mass is located closer to the axis of rotation, as this lowers its moment of inertia compared to a mass distributed farther from the axis.
This document summarizes key topics in linear motion including:
1) Linear motion concepts like position, displacement, velocity, and acceleration are discussed. Different reference systems are also introduced.
2) Equations for displacement, velocity, and acceleration are provided including how to calculate these values using derivatives of position over time.
3) Specific linear motion examples like uniform and non-uniform accelerated motion are examined along with example problems.
This document discusses rectilinear motion of particles. Rectilinear motion refers to motion along a straight line, with position defined by the distance x from a fixed origin O. Velocity is defined as the rate of change of position with respect to time. Acceleration is defined as the rate of change of velocity with respect to time. Displacement refers to the change in position, while distance traveled is the total length of the path traveled over a time interval. The document provides mathematical definitions and derivations for these key concepts in rectilinear motion.
The document describes circular uniform motion (MCU) and its parameters. MCU is a periodic motion where the body moves along a circular path with constant instantaneous speed. Key parameters include: period, frequency, tangential velocity, angular velocity, and centripetal acceleration. An example problem is given to calculate these values for a coin rotating on a spinning disk. Forces acting on the coin include its weight, the normal force, and static friction, with the latter providing the necessary centripetal force.
DINAMIKA ROTASI DAN KESETIMBANGAN BENDA TEGARmateripptgc
ย
1. The document discusses concepts in rotational dynamics including torque, moment of inertia, conservation of angular momentum, rotational kinetic energy, and rolling motion. It provides definitions, formulas, and example problems.
2. Key concepts covered include torque as a tendency of a force to cause rotation, moment of inertia as a measure of an object's resistance to changes in its rotation, and relationships between torque, moment of inertia, angular acceleration, angular momentum, and rotational kinetic energy.
3. Example problems calculate values like torque, moment of inertia, angular momentum, rotational kinetic energy, linear and angular velocities, and accelerations in rolling and rotational systems.
This document discusses rotational motion and provides definitions and equations for key angular quantities such as angular displacement (ฮธ), angular velocity (ฯ), angular acceleration (ฮฑ), torque (ฯ), moment of inertia (I), angular momentum (L), and rotational kinetic energy. It defines these quantities, gives their relationships to linear motion quantities, and provides examples of how to set up and solve problems involving rotational dynamics.
This document discusses circular motion and its key components. It covers:
1) Circular motion involves an object moving along a circular path, which can be described using angular position, velocity, and acceleration instead of linear measurements.
2) The relationships between angular and linear measurements are defined, such as how angular velocity relates to tangential linear velocity.
3) Uniform circular motion and uniformly accelerated circular motion are analyzed, with equations provided for how to calculate variables like displacement given velocity or acceleration.
4) The components of acceleration are described as normal (perpendicular to the path) and tangential, with equations for each in terms of angular acceleration.
This document discusses rotational motion and related concepts. It defines angular quantities like angular displacement, velocity, and acceleration and explains how they relate to linear motion. Torque is introduced as the product of force and lever arm that produces rotational acceleration. Rotational inertia, the resistance of an object to changes in its rotation, is defined. Examples show how to calculate angular and linear velocities/accelerations for objects in rotational motion.
Componentes tangenciales y normales diapositivas ostaiza nicoleNicoleOstaiza
ย
This document describes a student project to design and build a model that demonstrates tangential and normal components of motion. The objectives are to learn about tangential and normal components, see how they work in a model, and learn about the importance of kinematics. The project will use materials like wood, nails, glue, and a spring to build the model. The document provides background information on curved motion, tangential and normal acceleration components, and radius of curvature. It concludes with recommendations for building the model and a list of references.
System Of Particles And Rotational MotionAkrita Kaur
ย
This document defines key terms and concepts related to rotational motion and systems of particles, including:
- Angular position, displacement, velocity, and acceleration
- Equations of rotational motion
- Moment of inertia and its calculation for different objects
- Parallel and perpendicular axis theorems for calculating moment of inertia
- Torque, angular momentum, and their relationship to moment of inertia and angular acceleration
- Conservation of angular momentum for systems with no external torque
Learn Online Courses of Subject Engineering Mechanics of First Year Engineering. Clear the Concepts of Engineering Mechanics Through Video Lectures and PDF Notes. Visit us: https://ekeeda.com/streamdetails/subject/Engineering-Mechanics
This document provides an overview of planar kinematics of rigid body motion. It describes three types of planar rigid body motion: translation, rotation about a fixed axis, and general plane motion. Translation can be rectilinear or curvilinear. Rotation about a fixed axis involves angular position, velocity, acceleration, and the motion of a point on the rotating body. General plane motion is a combination of translation and rotation. Formulas are provided for analyzing velocity and acceleration during these different types of motion. Examples are also given to demonstrate how to apply the kinematic equations.
The document summarizes key concepts in rotational motion, including:
1) Torque is defined as the force applied tangentially to an object's axis of rotation, and is proportional to the lever arm and perpendicular force.
2) Static equilibrium occurs when the net torque on a system is zero, meaning torques cancel out.
3) For objects experiencing angular acceleration, net torque is related to angular acceleration by an angular analogue of Newton's Second Law.
The document discusses rotational motion and kinematics. It defines key concepts like the radian, angular velocity, and angular acceleration. It describes how to relate linear and rotational motion through equations. It also introduces the concept of moment of inertia, which describes an object's resistance to changes in rotational motion based on its mass distribution. Different formulas are given for calculating the moment of inertia of objects like rods, disks, and point masses rotating around different axes.
This document discusses uniform circular motion and related concepts like centripetal acceleration and centripetal force. It covers topics like how radius, speed and acceleration are related in uniform circular motion; the direction of velocity and acceleration vectors; forces that cause an object to travel in a circular path like friction or the normal force on a banked curve; and applications involving objects moving in horizontal and vertical circles like cars on curved roads. The document contains learning objectives, definitions, examples, questions and sections on key ideas like centripetal acceleration, centripetal force and banked curves.
Kinematics of a Particle document discusses:
1) Kinematics involves describing motion without considering forces, studying how position, velocity, and acceleration change over time for a particle.
2) Rectilinear motion involves a particle moving along a straight line, where position (x) is defined as the distance from a fixed origin, velocity (v) is the rate of change of position over time, and acceleration (a) is the rate of change of velocity over time.
3) Examples are provided to demonstrate solving kinematics problems using differentiation, integration, and relationships between position, velocity, acceleration graphs. Problems involve determining velocity, acceleration, distance or displacement given various relationships between these quantities.
This document discusses uniformly accelerated or uniformly varied rectilinear motion. It explains that in uniformly varied rectilinear motion, there is a change in velocity and therefore an acceleration. It defines acceleration as the rate of change of velocity. It provides the equation to calculate acceleration using the change in velocity over time. It also notes that the acceleration on a particle is constant since the mass is constant.
This document discusses analyzing the motion of particle systems using Newton's laws of motion. It defines a particle as a point mass with no orientation or rotational inertia, and discusses describing particle position, velocity, and acceleration using Cartesian components of position, velocity, and acceleration vectors. It presents Newton's three laws of motion and provides everyday examples. It also discusses calculating forces required to cause prescribed particle motions using free body diagrams and Newton's second law, and deriving and solving equations of motion for particle systems.
This document discusses kinematics of rigid bodies, including:
- Types of rigid body motion such as translation, rotation about a fixed axis, and general plane motion.
- Translation motion is further divided into rectilinear and curvilinear types.
- Key terms related to rotation about a fixed axis like angular position, displacement, velocity, and acceleration.
- Relations between linear and angular velocity and acceleration.
- Two special cases involving rotation of pulleys - a pulley connected to a rotating block, and two coupled pulleys rotating without slip.
- Five sample problems calculating values like angular velocity and acceleration, revolutions, linear velocity and acceleration for rotating bodies.
This document discusses uniform circular motion and related concepts. It begins by defining uniform circular motion as motion at constant speed in a circular path. It then derives the formula for centripetal acceleration and explains that a centripetal force is needed to provide the acceleration toward the center required for circular motion. Examples are provided to illustrate calculating centripetal force for different objects in circular motion, including effects of speed and radius. The document also discusses banked curves and satellites in circular orbits, providing the relevant equations and example calculations.
This document provides an overview of key concepts in rotational kinematics covered in Chapter 8, including angular displacement, velocity, and acceleration. It defines these rotational variables and their relationships to linear motion. Examples are given to illustrate calculating angular variables and transforming between rotational and tangential linear motion for objects like rolling wheels or helicopter blades. Formulas for rotational kinematics with constant angular acceleration are also presented.
This document discusses forces and Newton's laws of motion. It begins by asking what causes an object to remain at rest or in motion, and defines force as a vector quantity that can change an object's motion. It then introduces Newton's three laws of motion: 1) An object remains at rest or in uniform motion unless acted upon by an external force, 2) The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass, and 3) For every action there is an equal and opposite reaction. Examples of different types of forces like normal force, friction, and weight are also provided.
This document provides an overview of kinematics concepts including displacement, speed, velocity, acceleration, and equations of motion. Key points covered include:
- Kinematics deals with describing motion without considering causes of motion like forces.
- Displacement, speed, velocity, and acceleration are defined. Equations of motion that relate these variables for constant acceleration are presented.
- Position-time and velocity-time graphs are introduced as ways to represent motion. The slope and area under graphs relate to velocity and displacement.
- Free fall near the Earth's surface provides a specific example where acceleration due to gravity is constant.
- Graphical analysis techniques are described for determining acceleration from velocity-time graphs.
This document contains instructions for a physics lab experiment on moment of inertia. The experiment has two parts:
Part I measures the moment of inertia of a disk by applying a torque from a hanging mass and measuring the angular acceleration.
Part II measures the moment of inertia of a rod with two movable masses by varying the mass positions and amounts and again measuring angular acceleration from a hanging torque source. Equations are provided to calculate moment of inertia from experimental measurements.
Physics 161Static Equilibrium and Rotational Balance Intro.docxrandymartin91030
ย
Physics 161
Static Equilibrium and Rotational Balance
Introduction
In Part I of this lab, you will observe static equilibrium for a meter stick suspended horizontally. In Part II, you will observe the rotational balance of a cylinder on an incline. You will vary the mass hanging from the side of the cylinder for different angles.
Reference
Young and Freedman, University Physics, 12th Edition: Chapter 11, section 3
Theory
Part I: When forces act on an extended body, rotations about axes on the body can result as well as translational motion from unbalanced forces. Static equilibrium occurs when the net force and the net torque are both equal to zero. We will examine a special case where forces are only acting in the vertical direction and can therefore be summed simply without breaking them into components:
(1)
Torques may be calculated about the axis of your choosing:
(2)
where torque is specified by the equation:
(3)
where d is the lever arm (or moment arm) for the force. The lever arm is the perpendicular distance from the line of force to the axis about which you are calculating the torque.
Normally, up is "+" and down is "-" for forces. For torques, it is convenient to define clockwise as "-" and counterclockwise as "+". Whatever you decide to do, be consistent with your signs and make sure you understand what a "+" or "-" value for your force or torque means directionally.
Part II: Any round object when placed on an incline has tendency of rotating towards the bottom of an incline. If the downward force that causes the object to accelerate down the slope is canceled by another force, the object will remain stationary on the incline. Figure 1 shows the configuration of the setup. In order to have the rubber cylinder in static equilibrium we should satisfy the following conditions:
(4)
Figure 1: Experimental setup for Part II
The condition that the net force along the x-axis (which is conveniently taken along the incline) must be zero yields the relationship. (Prove this!)
Without static friction the cylinder would slide down the incline; the presence of friction causes a torque in clockwise (negative) direction. In order to have static equilibrium we need to balance that torque with a torque in counterclockwise direction. This is achieved by hanging the appropriate mass m.
Applying the last condition to the center of the cylinder will result in:
where r, the radius of the small cylinder (PVC fitting), is the moment arm for the mass m and R, the radius of the rubber cylinder, is the moment arm for the frictional force which accounts for M and m. Combining this equation with the equation for Ffr from above will result in:
(5)
(6)
By adjusting the mass m, we can observe how the equilibrium can be achieved.
Procedure
Part I: Static Equilibrium
Figure 2: Diagram of Torque Experiment Setup
1. Weigh the meter stick you use, including the metal hangers.
2. Attach .
The document describes circular uniform motion (MCU) and its parameters. MCU is a periodic motion where the body moves along a circular path with constant instantaneous speed. Key parameters include: period, frequency, tangential velocity, angular velocity, and centripetal acceleration. An example problem is given to calculate these values for a coin rotating on a spinning disk. Forces acting on the coin include its weight, the normal force, and static friction, with the latter providing the necessary centripetal force.
DINAMIKA ROTASI DAN KESETIMBANGAN BENDA TEGARmateripptgc
ย
1. The document discusses concepts in rotational dynamics including torque, moment of inertia, conservation of angular momentum, rotational kinetic energy, and rolling motion. It provides definitions, formulas, and example problems.
2. Key concepts covered include torque as a tendency of a force to cause rotation, moment of inertia as a measure of an object's resistance to changes in its rotation, and relationships between torque, moment of inertia, angular acceleration, angular momentum, and rotational kinetic energy.
3. Example problems calculate values like torque, moment of inertia, angular momentum, rotational kinetic energy, linear and angular velocities, and accelerations in rolling and rotational systems.
This document discusses rotational motion and provides definitions and equations for key angular quantities such as angular displacement (ฮธ), angular velocity (ฯ), angular acceleration (ฮฑ), torque (ฯ), moment of inertia (I), angular momentum (L), and rotational kinetic energy. It defines these quantities, gives their relationships to linear motion quantities, and provides examples of how to set up and solve problems involving rotational dynamics.
This document discusses circular motion and its key components. It covers:
1) Circular motion involves an object moving along a circular path, which can be described using angular position, velocity, and acceleration instead of linear measurements.
2) The relationships between angular and linear measurements are defined, such as how angular velocity relates to tangential linear velocity.
3) Uniform circular motion and uniformly accelerated circular motion are analyzed, with equations provided for how to calculate variables like displacement given velocity or acceleration.
4) The components of acceleration are described as normal (perpendicular to the path) and tangential, with equations for each in terms of angular acceleration.
This document discusses rotational motion and related concepts. It defines angular quantities like angular displacement, velocity, and acceleration and explains how they relate to linear motion. Torque is introduced as the product of force and lever arm that produces rotational acceleration. Rotational inertia, the resistance of an object to changes in its rotation, is defined. Examples show how to calculate angular and linear velocities/accelerations for objects in rotational motion.
Componentes tangenciales y normales diapositivas ostaiza nicoleNicoleOstaiza
ย
This document describes a student project to design and build a model that demonstrates tangential and normal components of motion. The objectives are to learn about tangential and normal components, see how they work in a model, and learn about the importance of kinematics. The project will use materials like wood, nails, glue, and a spring to build the model. The document provides background information on curved motion, tangential and normal acceleration components, and radius of curvature. It concludes with recommendations for building the model and a list of references.
System Of Particles And Rotational MotionAkrita Kaur
ย
This document defines key terms and concepts related to rotational motion and systems of particles, including:
- Angular position, displacement, velocity, and acceleration
- Equations of rotational motion
- Moment of inertia and its calculation for different objects
- Parallel and perpendicular axis theorems for calculating moment of inertia
- Torque, angular momentum, and their relationship to moment of inertia and angular acceleration
- Conservation of angular momentum for systems with no external torque
Learn Online Courses of Subject Engineering Mechanics of First Year Engineering. Clear the Concepts of Engineering Mechanics Through Video Lectures and PDF Notes. Visit us: https://ekeeda.com/streamdetails/subject/Engineering-Mechanics
This document provides an overview of planar kinematics of rigid body motion. It describes three types of planar rigid body motion: translation, rotation about a fixed axis, and general plane motion. Translation can be rectilinear or curvilinear. Rotation about a fixed axis involves angular position, velocity, acceleration, and the motion of a point on the rotating body. General plane motion is a combination of translation and rotation. Formulas are provided for analyzing velocity and acceleration during these different types of motion. Examples are also given to demonstrate how to apply the kinematic equations.
The document summarizes key concepts in rotational motion, including:
1) Torque is defined as the force applied tangentially to an object's axis of rotation, and is proportional to the lever arm and perpendicular force.
2) Static equilibrium occurs when the net torque on a system is zero, meaning torques cancel out.
3) For objects experiencing angular acceleration, net torque is related to angular acceleration by an angular analogue of Newton's Second Law.
The document discusses rotational motion and kinematics. It defines key concepts like the radian, angular velocity, and angular acceleration. It describes how to relate linear and rotational motion through equations. It also introduces the concept of moment of inertia, which describes an object's resistance to changes in rotational motion based on its mass distribution. Different formulas are given for calculating the moment of inertia of objects like rods, disks, and point masses rotating around different axes.
This document discusses uniform circular motion and related concepts like centripetal acceleration and centripetal force. It covers topics like how radius, speed and acceleration are related in uniform circular motion; the direction of velocity and acceleration vectors; forces that cause an object to travel in a circular path like friction or the normal force on a banked curve; and applications involving objects moving in horizontal and vertical circles like cars on curved roads. The document contains learning objectives, definitions, examples, questions and sections on key ideas like centripetal acceleration, centripetal force and banked curves.
Kinematics of a Particle document discusses:
1) Kinematics involves describing motion without considering forces, studying how position, velocity, and acceleration change over time for a particle.
2) Rectilinear motion involves a particle moving along a straight line, where position (x) is defined as the distance from a fixed origin, velocity (v) is the rate of change of position over time, and acceleration (a) is the rate of change of velocity over time.
3) Examples are provided to demonstrate solving kinematics problems using differentiation, integration, and relationships between position, velocity, acceleration graphs. Problems involve determining velocity, acceleration, distance or displacement given various relationships between these quantities.
This document discusses uniformly accelerated or uniformly varied rectilinear motion. It explains that in uniformly varied rectilinear motion, there is a change in velocity and therefore an acceleration. It defines acceleration as the rate of change of velocity. It provides the equation to calculate acceleration using the change in velocity over time. It also notes that the acceleration on a particle is constant since the mass is constant.
This document discusses analyzing the motion of particle systems using Newton's laws of motion. It defines a particle as a point mass with no orientation or rotational inertia, and discusses describing particle position, velocity, and acceleration using Cartesian components of position, velocity, and acceleration vectors. It presents Newton's three laws of motion and provides everyday examples. It also discusses calculating forces required to cause prescribed particle motions using free body diagrams and Newton's second law, and deriving and solving equations of motion for particle systems.
This document discusses kinematics of rigid bodies, including:
- Types of rigid body motion such as translation, rotation about a fixed axis, and general plane motion.
- Translation motion is further divided into rectilinear and curvilinear types.
- Key terms related to rotation about a fixed axis like angular position, displacement, velocity, and acceleration.
- Relations between linear and angular velocity and acceleration.
- Two special cases involving rotation of pulleys - a pulley connected to a rotating block, and two coupled pulleys rotating without slip.
- Five sample problems calculating values like angular velocity and acceleration, revolutions, linear velocity and acceleration for rotating bodies.
This document discusses uniform circular motion and related concepts. It begins by defining uniform circular motion as motion at constant speed in a circular path. It then derives the formula for centripetal acceleration and explains that a centripetal force is needed to provide the acceleration toward the center required for circular motion. Examples are provided to illustrate calculating centripetal force for different objects in circular motion, including effects of speed and radius. The document also discusses banked curves and satellites in circular orbits, providing the relevant equations and example calculations.
This document provides an overview of key concepts in rotational kinematics covered in Chapter 8, including angular displacement, velocity, and acceleration. It defines these rotational variables and their relationships to linear motion. Examples are given to illustrate calculating angular variables and transforming between rotational and tangential linear motion for objects like rolling wheels or helicopter blades. Formulas for rotational kinematics with constant angular acceleration are also presented.
This document discusses forces and Newton's laws of motion. It begins by asking what causes an object to remain at rest or in motion, and defines force as a vector quantity that can change an object's motion. It then introduces Newton's three laws of motion: 1) An object remains at rest or in uniform motion unless acted upon by an external force, 2) The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass, and 3) For every action there is an equal and opposite reaction. Examples of different types of forces like normal force, friction, and weight are also provided.
This document provides an overview of kinematics concepts including displacement, speed, velocity, acceleration, and equations of motion. Key points covered include:
- Kinematics deals with describing motion without considering causes of motion like forces.
- Displacement, speed, velocity, and acceleration are defined. Equations of motion that relate these variables for constant acceleration are presented.
- Position-time and velocity-time graphs are introduced as ways to represent motion. The slope and area under graphs relate to velocity and displacement.
- Free fall near the Earth's surface provides a specific example where acceleration due to gravity is constant.
- Graphical analysis techniques are described for determining acceleration from velocity-time graphs.
This document contains instructions for a physics lab experiment on moment of inertia. The experiment has two parts:
Part I measures the moment of inertia of a disk by applying a torque from a hanging mass and measuring the angular acceleration.
Part II measures the moment of inertia of a rod with two movable masses by varying the mass positions and amounts and again measuring angular acceleration from a hanging torque source. Equations are provided to calculate moment of inertia from experimental measurements.
Physics 161Static Equilibrium and Rotational Balance Intro.docxrandymartin91030
ย
Physics 161
Static Equilibrium and Rotational Balance
Introduction
In Part I of this lab, you will observe static equilibrium for a meter stick suspended horizontally. In Part II, you will observe the rotational balance of a cylinder on an incline. You will vary the mass hanging from the side of the cylinder for different angles.
Reference
Young and Freedman, University Physics, 12th Edition: Chapter 11, section 3
Theory
Part I: When forces act on an extended body, rotations about axes on the body can result as well as translational motion from unbalanced forces. Static equilibrium occurs when the net force and the net torque are both equal to zero. We will examine a special case where forces are only acting in the vertical direction and can therefore be summed simply without breaking them into components:
(1)
Torques may be calculated about the axis of your choosing:
(2)
where torque is specified by the equation:
(3)
where d is the lever arm (or moment arm) for the force. The lever arm is the perpendicular distance from the line of force to the axis about which you are calculating the torque.
Normally, up is "+" and down is "-" for forces. For torques, it is convenient to define clockwise as "-" and counterclockwise as "+". Whatever you decide to do, be consistent with your signs and make sure you understand what a "+" or "-" value for your force or torque means directionally.
Part II: Any round object when placed on an incline has tendency of rotating towards the bottom of an incline. If the downward force that causes the object to accelerate down the slope is canceled by another force, the object will remain stationary on the incline. Figure 1 shows the configuration of the setup. In order to have the rubber cylinder in static equilibrium we should satisfy the following conditions:
(4)
Figure 1: Experimental setup for Part II
The condition that the net force along the x-axis (which is conveniently taken along the incline) must be zero yields the relationship. (Prove this!)
Without static friction the cylinder would slide down the incline; the presence of friction causes a torque in clockwise (negative) direction. In order to have static equilibrium we need to balance that torque with a torque in counterclockwise direction. This is achieved by hanging the appropriate mass m.
Applying the last condition to the center of the cylinder will result in:
where r, the radius of the small cylinder (PVC fitting), is the moment arm for the mass m and R, the radius of the rubber cylinder, is the moment arm for the frictional force which accounts for M and m. Combining this equation with the equation for Ffr from above will result in:
(5)
(6)
By adjusting the mass m, we can observe how the equilibrium can be achieved.
Procedure
Part I: Static Equilibrium
Figure 2: Diagram of Torque Experiment Setup
1. Weigh the meter stick you use, including the metal hangers.
2. Attach .
En la temรกtica de hoy, hablaremos sobre como calcular los fenรณmenos en una maqueta de dinรกmica rotacional de dos objetos y saber cuanto se elonga los tres resortes.
1 Lab 3 Newtonโs Second Law of Motion Introducti.docxmercysuttle
ย
1
Lab 3: Newtonโs Second Law of Motion
Introduction
Newtonโs Second law of motion can be summarized by the following equation:
ฮฃ F = m a (1)
where ฮฃ F represents a net force acting on an object, m is the mass of the object moving
under the influence of ฮฃ F, and a is the acceleration of that object. The bold letters in
the equation represent vector quantities.
In this lab you will try to validate this law by applying Eq. 1 to the almost frictionless
motion of a car moving along a horizontal aluminum track when a constant force T
(tension in the string) acts upon it. This motion (to be exact the velocity of the moving
object) will be recorded automatically by a motion sensor. The experimental set up
for a car moving away from the motion sensor is depicted below.
If we consider the frictionless motion of the cart in the positive x-direction chosen in
the diagram, then Newtonโs Second Law can be written for each of the objects as
follows:
T Ma๏ฝ (2)
and
โ gT F ma๏ฝ ๏ญ (3)
From this system of equations we can get the acceleration of the system:
2
gF
a
m M
๏ฝ
๏ซ
(4)
Because the motion of the car is not frictionless, to get better results it is necessary to
include the force of kinetic friction fk experienced by the moving car in the analysis.
When the cart is moving away from the motion detector (positive x-direction in the
diagram) Newtonโs Second Law is written as follows for each of the moving objects
m and M:
1 1โ kT f Ma๏ฝ (5)
and
1 1โ gT F ma๏ฝ ๏ญ (6)
Since it is quite difficult to assess quantitatively the magnitude of kinetic friction
involved in our experiment we will solve the problem by putting the object in two
different situations in which the friction acts in opposite directions respectively while
the tension in the string remains the same.
When the cart M is forced to move towards the motion detector (negative x-direction
in the diagram), the corresponding Newtonโs Second Law equations will change as
follows:
2 2kT f Ma๏ซ ๏ฝ ๏ญ (7)
and
2 2gT F ma๏ญ ๏ฝ (8)
Note that in equations 5, 6, 7, and 8 the direction of acceleration represented by vector
a has been chosen in the same direction as the direction of motion.
We are able to eliminate the force of kinetic friction on the final result, by calculating
the mean acceleration from these two runs:
๏จ ๏ฉ1 2
2
ave
slope slope
a
๏ซ
๏ฝ (9)
Combing the equations (5) โ (8) we derive the equation to calculate the value of
gravitational acceleration:
๏จ ๏ฉavea M mg
m
๏ซ
๏ฝ (10)
3
Equipment
Horizontal dynamics track with smart pulley and safety stopper on one end; collision
cart with reflector connected to a variable mass hanging over the pulley; motion
detector connected to the Science Workshop interface recording the velocity of the
moving cart.
Procedure:
a) Weigh the cart (M) and the small mass (m) hanger.
b) Open the experiment file โNew ...
Simple harmonic movement in bandul reversibelumammuhammad27
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This document summarizes a practicum report on simple harmonic motion using a reversible pendulum. The objectives were to determine how factors like gravitational acceleration are affected in a reversible pendulum, compare measured and theoretical gravitational acceleration, determine the gravitational constant using the pendulum, and measure the pendulum's period of oscillation. Two experiments were conducted with the load in different positions, and period and gravitational acceleration were calculated using the measured time for oscillations. The measured gravitational accelerations were close to the theoretical value, with relative errors between 0.2-12.4%.
1 February 28, 2016 Dr. Samuel Daniels Associate.docxoswald1horne84988
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This report summarizes the procedures and results of an impact force lab experiment. The lab setup included wiring a strain gauge in a half-bridge configuration and using LabView to program sensors and collect data. Data was collected at angle increments of 5 degrees from 5 to 120 degrees and converted from strain to force. The experimental force values followed a sinusoidal trend when plotted against angle. The natural frequency was calculated and compared to the period of oscillation determined from raw waveform graphs, showing similar values between theoretical and experimental results. Some sources of error are noted, including noise in the raw waveform graphs and an incomplete angle range for the data.
This document describes an experiment to determine the moment of inertia of a flywheel both with and without additional masses attached. Students measured the angular acceleration of the flywheel when subjected to different torques provided by hanging weights. From the measured angular accelerations and known torques, the moments of inertia of the flywheel assembly were calculated. The experimentally determined moments of inertia matched reasonably well with the theoretical calculations. The experiment allowed students to apply their understanding of rotational motion and moment of inertia.
This document provides instructions for a physics laboratory practice on determining centers of mass and moments of inertia. It includes definitions of key terms like center of gravity, center of mass, and formulas for calculating the center of mass of different shapes. The activities to be completed involve using these formulas and measurements to find the center of mass for various 3D objects, 2D shapes, curves and functions, and an object made of soap. Results will be recorded in tables and used to also determine moments of inertia for each object analyzed. The goal is to apply the analysis of shapes and formulas to obtain the center of gravity or mass point for different physical examples.
This document discusses fundamental concepts in engineering physics including SI units, statics, vectors, forces, moments, and couples. It defines scalar and vector quantities, explains the parallelogram law of forces and Lami's theorem for resolving forces. Examples are also provided to illustrate concepts like finding the resultant of two forces and calculating torque.
analyzing system of motion of a particlesvikasaucea
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This document discusses analyzing the motion of particle systems using Newton's laws of motion. It begins by defining a particle and describing the position, velocity, and acceleration vectors of a particle. It then discusses how to use Newton's laws to calculate the forces needed to cause a particle to move in a particular way and how to derive equations of motion for particle systems. Examples are provided on simple harmonic motion and calculating the forces required to tip over a bicycle. The document concludes by outlining the general procedure for deriving and solving equations of motion for systems of particles.
The document describes the process of reverse engineering and modeling a testing part consisting of a shaft with gears. Key steps included:
1. Measuring the part dimensions and determining gear parameters.
2. Developing a modeling strategy starting with a revolve feature to create the shaft profile, then adding holes, rounds, chamfers and gears.
3. Modeling the gear teeth using involute curves defined by gear standards and patterned axially according to the number of teeth.
4. Completing the part model and measuring properties like volume and mass for later experimental validation.
This document provides instructions for a physics laboratory experiment on centers of mass. The objectives are to experimentally determine the center of mass of various objects and compare the results to calculations using center of mass formulas. Students will use materials like clay, soap, and rulers to find the centers of mass. Background information is provided on calculating center of mass for discrete particle systems, continuous bodies, and conical solids using integrals and density. Safety recommendations and a list of required equipment are also included.
After reading this module, you should be able to . . .
10.01 Identify that if all parts of a body rotate around a fixed
axis locked together, the body is a rigid body. (This chapter
is about the motion of such bodies.)
10.02 Identify that the angular position of a rotating rigid body
is the angle that an internal reference line makes with a
fixed, external reference line.
10.03 Apply the relationship between angular displacement
and the initial and final angular positions.
10.04 Apply the relationship between average angular velocity, angular displacement, and the time interval for that displacement.
10.05 Apply the relationship between average angular acceleration, change in angular velocity, and the time interval for
that change.
10.06 Identify that counterclockwise motion is in the positive
direction and clockwise motion is in the negative direction.
10.07 Given angular position as a function of time, calculate the
instantaneous angular velocity at any particular time and the
average angular velocity between any two particular times.
10.08 Given a graph of angular position versus time, determine the instantaneous angular velocity at a particular time
and the average angular velocity between any two particular times.
10.09 Identify instantaneous angular speed as the magnitude
of the instantaneous angular velocity.
10.10 Given angular velocity as a function of time, calculate
the instantaneous angular acceleration at any particular
time and the average angular acceleration between any
two particular times.
10.11 Given a graph of angular velocity versus time, determine the instantaneous angular acceleration at any particular time and the average angular acceleration between
any two particular times.
10.12 Calculate a bodyโs change in angular velocity by
integrating its angular acceleration function with respect
to time.
10.13 Calculate a bodyโs change in angular position by integrating its angular velocity function with respect to time.
The document discusses rotational motion and angular quantities. It defines angular displacement, velocity, and acceleration and describes how they relate to linear motion. It discusses torque as the product of force and lever arm that produces rotational acceleration. The moment of inertia depends on an object's mass distribution and axis of rotation, and determines the rotational acceleration produced by a torque. Rotational kinetic energy is defined analogously to linear kinetic energy, using angular quantities rather than linear ones.
This document provides instructions for a physics laboratory experiment on rotational dynamics. The objectives are to learn about rotational dynamics, identify inertia in different bodies, study how bodies behave in a model, and relate equations used to solve rotational dynamics problems. It describes building a model for this and provides equations for torque, angular acceleration, moment of inertia, and angular momentum. The procedure involves designing the model, calculating translational and rotational energies, and analyzing a collision using rotational dynamics equations. Calculations are shown for velocities before and after a simulated collision.
REPORT SUMMARYVibration refers to a mechanical.docxdebishakespeare
ย
REPORT SUMMARY
Vibration refers to a mechanical phenomenon involving oscillations about a point. These oscillations can be of any imaginable range of amplitudes and frequencies, with each combination having its own effect. These effects can be positive and purposefully induced, but they can also be unintentional and catastrophic. It's therefore imperative to understand how to classify and model vibration.
Within the classroom portion of ME 345, we discussed damped and undamped vibrations, appropriate models, and several of their properties. The purpose of Lab 3 is to give us the corresponding "hands-on" experience to cement our understanding of the theory.
As it turns out, vibration can be modeled with a simple spring-mass system (spring-mass-damper system for damped vibration). In order to create a mathematical model for our simple spring-mass system, we apply Newton's second law and sum the forces about the mass. After applying some of our knowledge of differential equations, the result is a second order linear differential equation (in vector form). This can easily be converted to the scalar version, from which it's easy to glean various properties of the vibration (i.e. natural frequency, period, etc.).
In the lab, we were provided with a PASCO motion sensor, USB link, ramp, and accompanying software. All of the aforementioned equipment was already assembled and connected. The ramp was set up at an angle with a stop on the elevated end and the motion sensor on the lower end. The sensor was connected to the USB link, which was in turn connected to the computer. We chose to use the Xplorer GLX software to interface with the sensor and record our data. After receiving our equipment, we gathered data on our spring's extension with a known load to derive a spring constant. We were provided with a small cart to which we attached weights to increase its mass. In order to model free vibration, we placed the cart on the track and attached it to the stop at the top of the ramp with a spring. After displacing the cart a certain distance from its equilibrium point, the cart was released and was allowed to oscillate on the track while we recorded its distance from the sensor. This was done with displacements of -20cm, -10cm, +10cm, and +20cm from the system's equilibrium point. After gathering this data for the "free" case, a magnet was attached to the front of the car, spaced as far from the track as possible. As the track is magnetic, this caused a slight damping effect, basically converting our spring-mass system to an underdamped spring-mass-damper system. After repeating the procedure for the "free" case, we moved the magnets as close to the track as possible (causing the system to become overdamped) and again repeated the procedure for the "free" case.
We were finally able to determine the period, phase angle, damping coefficients, and circular and cyclical frequencies for the three systems. There were similarities and differ ...
The document reports on a practicum about moment of inertia. It details the objectives, which were to determine period, moment of inertia, and deviation for various objects. It describes the theory behind moment of inertia and how it is analogous to mass for rotational motion. Tables show the tools used and experimental steps taken to collect data on mass, diameters, heights, periods, and deviations of different objects. The data collected is presented and calculations are shown for moment of inertia of various standard shapes.
This paper presents the Physics Rotational Method of the simple gravity pendulum, and it also applies Physics Direct Method to represent these equations, in addition to the numerical solutions discusses. This research investigates the relationship between angular acceleration and angle to find out different numerical solution by using simulation to see their behavior which shows in last part of this article.
The document contains multiple physics problems related to rotation, torque, angular acceleration, and moments of inertia. One problem involves calculating the angular speed of a winch drum that is lifting a 2000 kg block at a constant speed of 8 cm/s. It is determined that:
- The tension in the cable equals 19.6 kN, given by the weight of the block
- The torque exerted on the winch drum by the cable is 5.9 mkN
- The angular speed of the winch drum is 0.27 rad/s
- The power required by the motor is 1.6 kW
A Numerical Integration Scheme For The Dynamic Motion Of Rigid Bodies Using T...IJRES Journal
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The dynamics of rigid bodies have been studied extensively. However, a certain class of time-integration schemes were not consistent since they added vectors not belonging to the same tangent space (so3), of the Lie group (SO3) of the Special Orthogonal transformations in E3. The work of Cardona[1,2], and later Makinen[3,4], highlighted this fact using the rotation vector as the main parameter in their derivations. Some other programs in multibody dynamics, such as the work of Haug[5], rely on the Euler parameters, instead of the rotation vector, as the main variable in their formulations. For this class of programs, different time-integration schemes could be used .This paper discusses one such a scheme. As an example of application, the spinning top was used in this paper. For such a problem, the approximate change of the potential energy was found to be an upper bound to the change in the actual total energy during a time step.
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CHINAโS GEO-ECONOMIC OUTREACH IN CENTRAL ASIAN COUNTRIES AND FUTURE PROSPECTjpsjournal1
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The rivalry between prominent international actors for dominance over Central Asia's hydrocarbon
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Use PyCharm for remote debugging of WSL on a Windo cf5c162d672e4e58b4dde5d797...shadow0702a
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This document serves as a comprehensive step-by-step guide on how to effectively use PyCharm for remote debugging of the Windows Subsystem for Linux (WSL) on a local Windows machine. It meticulously outlines several critical steps in the process, starting with the crucial task of enabling permissions, followed by the installation and configuration of WSL.
The guide then proceeds to explain how to set up the SSH service within the WSL environment, an integral part of the process. Alongside this, it also provides detailed instructions on how to modify the inbound rules of the Windows firewall to facilitate the process, ensuring that there are no connectivity issues that could potentially hinder the debugging process.
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Car accident rates have increased in recent years, resulting in losses in human lives, properties, and other financial costs. An embedded machine learning-based system is developed to address this critical issue. The system can monitor road conditions, detect driving patterns, and identify aggressive driving behaviors. The system is based on neural networks trained on a comprehensive dataset of driving events, driving styles, and road conditions. The system effectively detects potential risks and helps mitigate the frequency and impact of accidents. The primary goal is to ensure the safety of drivers and vehicles. Collecting data involved gathering information on three key road events: normal street and normal drive, speed bumps, circular yellow speed bumps, and three aggressive driving actions: sudden start, sudden stop, and sudden entry. The gathered data is processed and analyzed using a machine learning system designed for limited power and memory devices. The developed system resulted in 91.9% accuracy, 93.6% precision, and 92% recall. The achieved inference time on an Arduino Nano 33 BLE Sense with a 32-bit CPU running at 64 MHz is 34 ms and requires 2.6 kB peak RAM and 139.9 kB program flash memory, making it suitable for resource-constrained embedded systems.
The CBC machine is a common diagnostic tool used by doctors to measure a patient's red blood cell count, white blood cell count and platelet count. The machine uses a small sample of the patient's blood, which is then placed into special tubes and analyzed. The results of the analysis are then displayed on a screen for the doctor to review. The CBC machine is an important tool for diagnosing various conditions, such as anemia, infection and leukemia. It can also help to monitor a patient's response to treatment.
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1. 1
Ing.DiegoProaรฑoMolina
UNIVERSIDAD DE LAS FUERZAS ARMADAS
ESPE SEDE LATACUNGA
DEPARTAMENTO DE CIENCIAS EXACTAS
GUรA DE PRรCTICA DE LABORATORIO
CARRERA
CรDIGO DE LA
ASIGNATURA
NOMBRE DE LA ASIGNATURA
AUTOMOTRIZ______
ELECTROMECรNICA__
ELECTRรNICA_______
PETROQUรMICA______
MECATRรNICA_______
SOFTWARE_____X____
EXCT- MVU-50
EXCT- MVU-53
EXCT- MVU -52
EXCT- MVU- 51
EXCT- MVU-54
A 0001
Fรญsica I
NRC:_______4173_______
PRรCTICA
Nยฐ
LABORATORIO DE: LABORATORIO DE FรSICA
DURACIรN
(HORAS)
2 TEMA: Dinรกmica Rotacional 2
1 OBJETIVO
Objetivo General:
๏ท Construir una maqueta de dinรกmica rotacional.
Objetivos Especรญficos:
๏ท Analizar los tiempos que la esfera y la partรญcula se mueven por el sistema.
๏ท Determinar los coeficientes de rozamiento de tres diferentes materiales.
๏ท Comprobar que tipo de choque se genera en la colisiรณn entre la esfera y la barra.
๏ท Estimar el grado de validez de una prรกctica de laboratorio (5%).
2
INSTRUCCIONES:
PRรSTAMO DE MATERIALES Y EQUIPAMIENTO
A. El Jefe del Laboratorio es el responsable del prรฉstamo de equipos,
B. El docente es el responsable de la supervisiรณn en el Laboratorio y guiado de los alumnos en el uso de ciertos equipos o
instrumentos.
C. El material del Laboratorio sรณlo podrรก ser utilizado por los usuarios inscritos en los cursos asociados alLaboratorio.
D. El material del Laboratorio sรณlo podrรก ser utilizado en el Laboratorio.
E. El usuario deberรก entregar su credencialde alumno para el prรฉstamo de materiales y firmar la hoja de prรฉstamo.
DAรOS A LOS MATERIALES Y EQUIPAMIENTO
A. El daรฑo o pรฉrdida del material en prรฉstamo es de entera responsabilidad de los usuarios (alumnos y/o investigadores) que
hayan solicitado el material prestado.
B. Los usuarios deberรกn pagar la reposiciรณn del material que solicitaron en caso que รฉste sea perdido o daรฑado
RECOMENDACIONES DE SEGURIDAD:
A. Revisar todos los equipos y materiales entregados para evitar malos entendidos por pรฉrdidas o daรฑos causados.
2. 2
Ing.DiegoProaรฑoMolina
B. Adecรบe su puesto de trabajo, retirando y ordenando todos los elementos que no sean utilizados o estorben en el lugar.
C. Revise que los equipos de mediciรณn no estรฉn averiados y se puedan encerar.
D. Evite golpear o dejar caer los elementos ya que sufrirรกn daรฑos y deberรกn ser reemplazados por quien lo haya averiado.
E. Controle su zona de trabajo para que no afecte su labor o la de sus compaรฑeros.
A. EQUIPO Y MATERIALES NECESARIOS
Tabla 1. Equipos y materiales de la prรกctica
Material Caracterรญsticas Cantidad Cรณdigo
a)
Calibrador Vernier
Precisiรณn 1/128 in, Mรกxima
medida es de 16 cm
1 VER-6PX
b)
Flexรณmetro
Precisiรณn 1mmm; medida
mรกxima es de 3m
1 PRO-3MEB
c) Masa de prueba Pesos 5
Incluida en la
Balanza mecรกnica
d) Resortes 3 N/A
e) Tablas Madera 10 N/A
f) Cristal 1 N/A
g) Iman Bola 1 N/A
h) Vela 1 N/A
i) Balanza Digital N/A
j) Cartรณn 1 N/A
Figura Nยฐ 1 Navarrete J(2021)
B. TRABAJO PREPARATORIO:
2.1 Definiciรณn de dinรกmica rotacional.
Cuando hablamos de dinรกmica rotacional hablamos de โCuando un objeto es sometido a una fuerza
ejercida a una cierta distancia de un origen O, el sistema adquirirรก una aceleraciรณn angular, debido
a la acciรณn de una cantidad fรญsica denominada torque.โ (รlvaro, A. 2015). Entonces entendemos
por dinรกmica rotacional, el estudio de las fuerzas que generan un movimiento en donde ocurre la
rotaciรณn el cual se define como โse considera fijo un punto, el รบnico movimiento posible es aquel en
el que cada uno de los otros puntos se mueve en la superficie de una esfera cuyo radio es la
distancia del punto mรณvil al punto fijo.โ (Cano, I. 2016)
3. 3
Ing.DiegoProaรฑoMolina
2.2 Momento de una fuerza o torque
El momento de una fuerza o torque se define como โse llama torque 0 momento de una fuerza a la
capacidad de dicha fuerza para producir un giro o rotaciรณn alrededor de un punto.โ (Lรณpez, M. 2016).
Entonces el torque puede ser comprendido como aquel fenรณmeno fรญsico que describe la rotaciรณn
de un objeto, es decir su fuerza generada, como se puede observar en la figura 1, el torque es lo
que define la rotaciรณn. Su unidad en el SI es el ๐ โ ๐ .Matematicamente Zamora, J. en 2018 define
al torque como:
Figura 2. Torque de una fuerza.
๐ = ๐น โ ๐ โ ๐ ๐๐(๐) (1)
Segรบn Mendoza, E. en 2015 el signo del momento de una fuerza se ve determinada por la direcciรณn
del giro, si dicho giro ocurre en sentido contrario al de las manecillas de un reloj el signo serรก positivo
y si sucede lo contrario, es decir, se mueve en sentido horario, este serรก negativo; aunque esto
puede cambiar ya que hay autores que toman en consideraciรณn el signo del momento de una fuerza
de manera contraria.
2.3 Condiciรณn de equilibrio de un sistema rotacional
Segรบn Santiago, A. y et al. En 2018 para considerar que un cuerpo no se encuentra rotando, es
decir, la sumatoria de su momento de la fuerza es igual a cero, se deben analizar dichos momentos
diferentes. Entonces entendemos que, aunque un cuerpo se encuentre en reposo en el eje de las
abscisas y en el eje de las ordenadas, este puede rotar independientemente, Beraha, N. en 2019
afirma que esto nos ayuda a llegar a la siguiente consideraciรณn al momento de realizar el anรกlisis
en la dinรกmica rotacional cuando esta estรก en reposo:
โ ๐ = 0 (2)
De acuerdo a lo expresado por Vargas, A. en 2017 cuando una partรญcula gira en torno a un punto
fijo por el efecto de fuerzas tangenciales, su aceleraci6n tangencial viene dada por la expresiรณn
dada en la fรณrmula matemรกtica 3. Esta fuerza tangencial podemos considerarla como nuestro
torque si multiplicamos dicha ecuaciรณn por la distancia que se tiene hasta el centro de rotaciรณn, de
esta forma obtenemos la formula descrita por Franco, A. en 2016:
โ ๐น๐ก = ๐ โ ๐ (3)
โ ๐ = โ ๐น๐ก โ ๐ = ๐ โ ๐ โ ๐ (4)
โ ๐ = ๐(๐ โ ๐) โ ๐ (5)
โ ๐ = ๐ โ ๐2 โ ๐ผ (6)
โ๐ = ๐ผ โ ๐ผ (7)
2.4 Teorema de Steiner o de ejes paralelos.
โEl momento de inercia respecto de un eje, es igual al momento de inercia respecto de un eje
paralelo y que pase por el centro de gravedad mรกs el producto de la masa por la distancia al
cuadrado entre ambos ejesโ (Llopis, J. 2018) Este teorema es รบtil ya que como dice su enunciado,
si conocemos el valor de un momento de inercia en un eje particular, podemos determinar el valor
de este en ejes paralelos de ahรญ el nombre. Cada forma geomรฉtrica al igual que en el caso del
centroide posee un valor para la inercia del mismo. Segรบn Vazquez, R. en 2020 la formula general
la tendrรญamos de la siguiente manera:
4. 4
Ing.DiegoProaรฑoMolina
๐ผ๐ฅ = ๐ผ๐ฅ๐ + ๐๐ท๐ฅ๐ฅ๐
2 (8)
Figura 3. Teorema de Steiner
2.5 Momento de inercia
โEl momento de inercia serรก la suma individual de cada una de las masas mรญ que componen un
cuerpo multiplicado por la distancia al cuadrado ๐๐2
hacia el eje de rotaciรฉnโ (Valcarce, A, 2015).
Este momento de inercia basรกndonos en lo dicho por Llopis, J. en 2018 relata que tan difรญcil es
cambiar la velocidad de rotaciรณn de un sistema; es decir, de manera intuitiva sabremos que tratar
de cambiar la velocidad de algo que estรฉ mรกs cerca del eje de rotaciรณn serรก mรกs sencillo que
cambiar dicha velocidad desde los mรกs lejano de dicho cuerpo. Su fรณrmula segรบn Valcarce, A. en
2015 es:
๐ผ = โ ๐๐๐๐
2 (9)
๐ผ๐ฅ = โซ ๐ฅ2
๐๐ด
๐ด
(10)
๐ผ๐ฆ = โซ ๐ฆ2
๐๐ด
๐ด
(11)
2.6 Tabla de momentos de inercia de cuerpos geomรฉtricos planos comunes
Los momentos de inercia pueden ser generalizados segรบn su forma geomรฉtrica de acuerdo a
Llopis, J. en 2018. En la siguiente tabla se presenta los momentos de inercia de los objetos con
respecto a los ejes de las abscisas y de las ordenadas respectivamente, esto nos ayuda por el
simple motivo que ya con estos momentos de inercia conocidos se puede aplicar el teorema de
Steiner de la secciรณn 2.4 para hallar el momento de inercia en cualquier eje paralelo a los ejes de
โxโ y โyโ
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Ing.DiegoProaรฑoMolina
Figura 4. Tabla de Momentos de Inercia adaptada de Alvarez, E. 2018.
2.7 Relaciones entre fenรณmenos angularesy traslacionales
Seguin Valcarce, A. en 2016 podemos relacionar efectivamente las cantidades que se presentan
de
manera lineal con las que se presentan de manera angular, cuando nos referimos a cantidades
lineales, hacemos referencia a la velocidad tangencial, la aceleraci6n tangencial y el
desplazamiento; de la misma manera cuando hablamos de cantidades angulares hablamos de la
velocidad angular, la aceleraciรณn angular y el desplazamiento angular. Entonces estas relaciones
de acuerdo al mismo Valcarce, A lo tendrรญamos de la siguiente manera:
๐ = ๐ ๐ (12)
๐ฃ = ๐ ๐ (13)
๐ฃ =
๐๐
๐๐ก
= ๐
๐๐
๐๐ก
= ๐ ๐ (14)
๐๐ =
๐ฃ2
๐
= ๐ ๐2 (15)
๐๐ก =
๐๐ฃ
๐๐ก
= ๐
๐๐
๐๐ก
= ๐ ๐ผ (16)
6. 6
Ing.DiegoProaรฑoMolina
2.7 Energรญas de la rotaciรณn, trabajo y momento angular en la rotaciรณn
segรบn Olmo, M. en 2016 las energรญas al momento de existir una rotaciรณn difieren de aquellas en la
cual solo existe traslaciรณn, si bien es cierto puede existir ambas en un sistema; tambiรฉn
complementa con el hecho de que el trabajo de un sistema en rotaciรณn puede ser determinado a
partir del torque del mismo, A su vez Olmo describe un factor extremadamente importante como lo
es el momento angular, relacionรกndolo con el momento de inercia de la siguiente manera:
๐ธ๐๐ =
1
2
๐ โ ๐2 (17)
๐ = ๐๐ (18)
๐๐๐๐ก๐ = ๐๐๐๐ก๐๐ = ๐ผ๐ผ๐ (19)
๐ฟ
โ = ๐ผ โ ๐
โ
โ (20)
3 ACTIVIDADES A DESARROLLAR
Ensayo 1: Mediciรณn de valores para el cรกlculo de tiempos de bajada de la esfera y la
barra, la fuerza de rozamiento de cada material con respecto a la esfera y la barra,
ademรกs del cรกlculo de las constantes elรกsticas.
๏ท Definir las variables necesarias para determinar la aceleraciรณn como lo es el tiempo y la
longitud que recorriรณ la esfera y la barra.
๏ท Con un instrumento de mediciรณn identificar la longitud de cada uno de los materiales por el
cual va a ir todo el sistema.
๏ท Con ayuda del cronometro determinar el tiempo de bajada de la esfera por toda la rampa,
๏ท Despuรฉs de bajar de la rampa calcular el tiempo que demora hasta llegar a golpear la barra
๏ท Luego calcular el tiempo en que la barra cae hasta llegar al sistema de resorte.
๏ท Y tambiรฉn tomar el tiempo en que el resorte se deforma.
๏ท Observar si existe un movimiento por parte del sistema de resortes si este impulsa a la
barra se queda deformado sin moverse.
4 RESULTADOS OBTENIDOS
Datos:
Tabla de variables fรญsicas de la prรกctica
Tabla Nยฐ 2 Variables fรญsica
Parรกmetro fรญsico Dimensiรณn Sรญmbolo Unidades
Masa M Kg kg
Volumen ๐ฟ3
๐๐3 cm
7. 7
Ing.DiegoProaรฑoMolina
Tablas de datos
Ensayo 1: Mediciรณn de variables para el cรกlculo del tiempo para la esfera
Tabla Nยฐ 3 Variables del tiempo
Nยบ de
ejecuciones
Desplazamiento
X(m)
Tiempo
(s)
1 0.3010 0.53
2 0.3010 0.53
3 0.3010 0.54
4 0.3010 0.54
5 0.3010 0.54
6 0.3010 0.53
7 0.3010 0.53
8 0.3010 0.53
9 0.3010 0.54
10 0.3010 0.53
VALORES DE MEDICIรN PARA EL CรLCULO DE LA ACELERACION DE LA ESFERA
Cรกlculo del error absoluto:
1. Calcular la media aritmรฉtica
๐ฅฬ = โ
๐ฅ1 + ๐ฅ2 + ๐ฅ๐
๐
๐
๐=1
Tabla Nยฐ 4 Promedio de medidas
Tiempo ๐ฅฬ
0.53 0.53 0.534
0.53 0.53
0.54 0.53
0.54 0.54
0.54 0.53
2. Calcular el error absoluto
๐ธ๐๐๐ ๐ = |๐ฅฬ โ ๐ฅ๐|
Tabla Nยฐ5 Cรกlculo del Error Absoluto
tiempo ๐ฅฬ ๐ธ๐๐๐ ๐
0.53 0.534 0.004
0.53 0.534 0.004
9. 9
Ing.DiegoProaรฑoMolina
6. Calculo del rango de valores
Tabla Nยฐ9 Valores mรกx y mรญn
Tiempo
(๐
ฬ ยฑ ๐ฌ๐๐๐
ฬ ฬ ฬ ฬ ฬ )
๐๐๐๐ = (๐. ๐๐๐ โ ๐. ๐๐๐๐))
๐๐๐๐ = ๐. ๐๐๐๐
๐๐รก๐ฅ = (๐. ๐๐๐ + ๐.๐๐๐๐))
๐๐รก๐ฅ = 0.5388
Tabla Nยฐ 10: Validez de datos
Tiempo Valor mรญnimo Valor mรกximo Valores aceptables
0.53 0.5292 0.5388 Aceptado
0.53 0.5292 0.5388 Aceptado
0.54 0.5292 0.5388 Rechazado
0.54 0.5292 0.5388 Rechazado
0.54 0.5292 0.5388 Rechazado
0.53 0.5292 0.5388 Aceptado
0.53 0.5292 0.5388 Aceptado
0.53 0.5292 0.5388 Aceptado
0.54 0.5292 0.5388 Rechazado
0.53 0.5292 0.5388 Aceptado
Ensayo 2: Mediciรณn de variables para el cรกlculo de la aceleraciรณn para la esfera
๐ =
2โ๐ฅ
๐ก2
Tabla Nยฐ11 Datos del volumen
Nยบ de
ejecuciones
Desplazamiento
X(m)
Tiempo
(s)
Aceleraciรณn
(m/s*s)
1 0.3010 0.53 2.1431
2 0.3010 0.53 2.1431
3 0.3010 0.54 2.0645
4 0.3010 0.54 2.0645
5 0.3010 0.54 2.0645
6 0.3010 0.53 2.1431
7 0.3010 0.53 2.1431
8 0.3010 0.53 2.1431
9 0.3010 0.54 2.0645
10 0.3010 0.53 2.1431
10. 10
Ing.DiegoProaรฑoMolina
VALORES DE MEDICIรN PARA EL CรLCULO DE LA ACELERACION DE LA ESFERA
Cรกlculo del error absoluto:
7. Calcular la media aritmรฉtica
๐ฅฬ = โ
๐ฅ1 + ๐ฅ2 + ๐ฅ๐
๐
๐
๐=1
Tabla Nยฐ 12 Promedio de medidas
Largo ๐ฅฬ
2.1431 2.1431 2.1116
2.1431 2.1431
2.0645 2.1431
2.0645 2.0645
2.0645 2.1431
8. Calcular el error absoluto
๐ธ๐๐๐ ๐ = |๐ฅฬ โ ๐ฅ๐|
Tabla Nยฐ13 Cรกlculo del Error Absoluto
Aceleraciรณn ๐ฅฬ ๐ธ๐๐๐ ๐
2.1431 2.1116 0.0315
2.1431 2.1116 0.0315
2.0645 2.1116 0.0471
2.0645 2.1116 0.0471
2.0645 2.1116 0.0471
2.1431 2.1116 0.0315
2.1431 2.1116 0.0315
2.1431 2.1116 0.0315
2.0645 2.1116 0.0471
2.1431 2.1116 0.0315
9. Calcular error absoluto medio.
๐ธ๐๐๐
ฬ ฬ ฬ ฬ ฬ = โ
๐ธ๐๐๐ 1 + ๐ธ๐๐๐ 2+๐ธ๐๐๐ ๐
๐
๐
๐=1
Tabla Nยฐ14 Promedio del Error Absoluto
11. 11
Ing.DiegoProaรฑoMolina
๐ธ๐๐๐ ๐ (Aceleraciรณn) ๐ธ๐๐๐
ฬ ฬ ฬ ฬ ฬ
0.0315 0.0315 0.03774
0.0315 0.0315
0.0471 0.0315
0.0471 0.0471
0.0471 0.0315
10. Calcular el error relativo
Tabla Nยฐ15 Error Relativo
Largo
๐ธ๐ =
๐ธ๐๐๐
ฬ ฬ ฬ ฬ ฬ
๐ฅฬ
๐ธ๐ =
(0.0377)
(2.1116)
๐ธ๐ = 0.0178
11. Calcular el error porcentual
Tabla Nยฐ16 Error promedio
Aceleraciรณn
๐ฌ% = ๐ฌ๐ โ ๐๐๐%
๐ธ% = 0.0178 โ 100%
๐ธ% = 1.78 %
12. Calculo del rango de valores
Tabla Nยฐ17 Valores mรกx y mรญn
Largo
(๐
ฬ ยฑ ๐ฌ๐๐๐
ฬ ฬ ฬ ฬ ฬ )
๐๐๐๐ = (๐. ๐๐๐๐ โ ๐. ๐๐๐๐)
๐๐๐๐ = ๐. ๐๐๐๐
๐๐รก๐ฅ = (๐. ๐๐๐๐ + ๐.๐๐๐๐))
๐๐รก๐ฅ = 2.1493
Tabla Nยฐ 18: Validez de datos
Aceleraciรณn
Valor
mรญnimo
Valor
mรกximo
Valores
aceptables
2.1431 2.0739 2.1493 Aceptado
2.1431 2.0739 2.1493 Aceptado
2.0645 2.0739 2.1493 Rechazado
2.0645 2.0739 2.1493 Rechazado
2.0645 2.0739 2.1493 Rechazado
2.1431 2.0739 2.1493 Aceptado
12. 12
Ing.DiegoProaรฑoMolina
2.1431 2.0739 2.1493 Aceptado
2.1431 2.0739 2.1493 Aceptado
2.0645 2.0739 2.1493 Rechazado
2.1431 2.0739 2.1493 Aceptado
Ensayo 3: Mediciรณn de variables para el cรกlculo del tiempo para la esfera hasta llegar a colisionar
con la barra.
Tabla Nยฐ19 Datos del volumen
Nยบ de
ejecuciones
Desplazamiento
X(m)
Tiempo
(s)
1 0.1140 0.21
2 0.1140 0.22
3 0.1140 0.21
4 0.1140 0.22
5 0.1140 0.21
6 0.1140 0.21
7 0.1140 0.21
8 0.1140 0.21
9 0.1140 0.21
10 0.1140 0.21
VALORES DE MEDICIรN PARA EL CรLCULO DEL TIEMPO DE LA ESFERA.
Cรกlculo del error absoluto:
13. Calcular la media aritmรฉtica
๐ฅฬ = โ
๐ฅ1 + ๐ฅ2 + ๐ฅ๐
๐
๐
๐=1
Tabla Nยฐ 20 Promedio de medidas
Tiempo ๐ฅฬ
0.21 0.21 0.212
0.22 0.21
0.21 0.21
0.22 0.21
0.21 0.21
24. 24
Ing.DiegoProaรฑoMolina
๐ธ% = 0.0110 โ 100%
๐ธ% = 1.1 %
42. Calculo del rango de valores
Tabla Nยฐ57 Valores mรกx y mรญn
u
(๐
ฬ ยฑ ๐ฌ๐๐๐
ฬ ฬ ฬ ฬ ฬ )
๐๐๐๐ = (๐. ๐๐๐๐ โ ๐. ๐๐๐๐)
๐๐๐๐ = ๐. ๐๐๐๐
๐๐รก๐ฅ = (๐. ๐๐๐๐ + ๐. ๐๐๐๐))
๐๐รก๐ฅ = ๐. ๐๐๐๐
Tabla Nยฐ 58: Validez de datos
u
Valor
mรญnimo
Valor
mรกximo
Valores
aceptables
0.4109 0.4102 ๐. ๐๐๐๐ Aceptado
0.4109 0.4102 ๐. ๐๐๐๐ Aceptado
0.4206 0.4102 ๐. ๐๐๐๐ Aceptado
0.4206 0.4102 ๐. ๐๐๐๐ Aceptado
0.4206 0.4102 ๐. ๐๐๐๐ Aceptado
0.4109 0.4102 ๐. ๐๐๐๐ Aceptado
0.4109 0.4102 ๐. ๐๐๐๐ Aceptado
0.4109 0.4102 ๐. ๐๐๐๐ Aceptado
0.4206 0.4102 ๐. ๐๐๐๐ Aceptado
0.4109 0.4102 ๐. ๐๐๐๐ Aceptado
Ensayo 8 : CALCULO DE EL COEFICIENTE DE ROZAMIENTO 2
๐๐ฅ โ ๐น๐ = ๐ โ ๐
๐ โ ๐ โ ๐ข โ ๐ = ๐ โ ๐
Tabla Nยฐ59 Datos del volumen
Nยบ de
ejecuciones
Desplazamiento
X(m)
Tiempo
(s)
Aceleraciรณn
(m/s*s)
Peso kg U
1 0.1140 0.21 5.1700 0.018 0.4727
2 0.1140 0.22 4.7107 0.018 0.5195
3 0.1140 0.21 5.1700 0.018 0.4727
4 0.1140 0.22 4.7107 0.018 0.5195
5 0.1140 0.21 5.1700 0.018 0.4727
6 0.1140 0.21 5.1700 0.018 0.4727
7 0.1140 0.21 5.1700 0.018 0.4727
8 0.1140 0.21 5.1700 0.018 0.4727
9 0.1140 0.21 5.1700 0.018 0.4727
10 0.1140 0.21 5.1700 0.018 0.4727
25. 25
Ing.DiegoProaรฑoMolina
VALORES DE MEDICIรN PARA EL CรLCULO DE LA ACELERACION DE LA ESFERA
Cรกlculo del error absoluto:
43. Calcular la media aritmรฉtica
๐ฅฬ = โ
๐ฅ1 + ๐ฅ2 + ๐ฅ๐
๐
๐
๐=1
Tabla Nยฐ 60 Promedio de medidas
u ๐ฅฬ
0.4727 0.4727 0.4820
0.5195 0.4727
0.4727 0.4727
0.5195 0.4727
0.4727 0.4727
44. Calcular el error absoluto
๐ธ๐๐๐ ๐ = |๐ฅฬ โ ๐ฅ๐|
Tabla Nยฐ61 Cรกlculo del Error Absoluto
u ๐ฅฬ ๐ธ๐๐๐ ๐
0.4727 0.4820 0.0093
0.5195 0.4820 0.0375
0.4727 0.4820 0.0093
0.5195 0.4820 0.0375
0.4727 0.4820 0.0093
0.4727 0.4820 0.0093
0.4727 0.4820 0.0093
0.4727 0.4820 0.0093
0.4727 0.4820 0.0093
0.4727 0.4820 0.0093
45. Calcular error absoluto medio.
๐ธ๐๐๐
ฬ ฬ ฬ ฬ ฬ = โ
๐ธ๐๐๐ 1 + ๐ธ๐๐๐ 2+๐ธ๐๐๐ ๐
๐
๐
๐=1
Tabla Nยฐ62 Promedio del Error Absoluto
26. 26
Ing.DiegoProaรฑoMolina
๐ธ๐๐๐ ๐ (u) ๐ธ๐๐๐
ฬ ฬ ฬ ฬ ฬ
0.0093 0.0093 0.0149
0.0375 0.0093
0.0093 0.0093
0.0375 0.0093
0.0093 0.0093
46. Calcular el error relativo
Tabla Nยฐ63 Error Relativo
u
๐ธ๐ =
๐ธ๐๐๐
ฬ ฬ ฬ ฬ ฬ
๐ฅฬ
๐ธ๐ =
(0.0149)
(0.4820)
๐ธ๐ = 0.0309
47. Calcular el error porcentual
Tabla Nยฐ64 Error promedio
u
๐ฌ% = ๐ฌ๐ โ ๐๐๐%
๐ธ% = 0.0309 โ 100%
๐ธ% = 3.09 %
48. Calculo del rango de valores
Tabla Nยฐ65 Valores mรกx y mรญn
u
(๐
ฬ ยฑ ๐ฌ๐๐๐
ฬ ฬ ฬ ฬ ฬ )
๐๐๐๐ = (๐. ๐๐๐๐ โ ๐. ๐๐๐๐)
๐๐๐๐ = ๐. ๐๐๐๐
๐๐รก๐ฅ = (๐. ๐๐๐๐ + ๐. ๐๐๐๐))
๐๐รก๐ฅ = ๐. ๐๐๐๐
Tabla Nยฐ 66: Validez de datos
u
Valor
mรญnimo
Valor
mรกximo
Valores
aceptables
0.4727 0.4671 ๐. ๐๐๐๐ Aceptado
0.5195 0.4671 ๐. ๐๐๐๐ Rechazado
0.4727 0.4671 ๐. ๐๐๐๐ Aceptado
0.5195 0.4671 ๐. ๐๐๐๐ Rechazado
0.4727 0.4671 ๐. ๐๐๐๐ Aceptado
0.4727 0.4671 ๐. ๐๐๐๐ Aceptado
27. 27
Ing.DiegoProaรฑoMolina
0.4727 0.4671 ๐. ๐๐๐๐ Aceptado
0.4727 0.4671 ๐. ๐๐๐๐ Aceptado
0.4727 0.4671 ๐. ๐๐๐๐ Aceptado
0.4727 0.4671 ๐. ๐๐๐๐ Aceptado
Ensayo 10: CALCULO DE LA CONSTANTE DE ELASTICIDAD EN TORNO ALOS RESORTES 1
Y 2 QUE TIENEN LA MISMADIMENSION.
VALORES DE MEDICIรN PARA EL CรLCULO DE VOLUMEN
Calculo de la k del resorte
๐ =
๐๐ โ ๐
(๐๐ โ ๐๐)
Tabla Nยฐ 67 Calculo constante K
Masa
kg
0.022 kg 0.043 kg 0.64 kg 0.084 kg 0.088 kg
Li m 0.080 0.080 0.080 0.080 0.080
Lf m 0.0973 0.1137 0.1301 0.146 0.149
K N/m 12.4713 12.5133 12.5279 12.4816 12.5074
Cรกlculo del error absoluto:
49. Calcular la media aritmรฉtica
๐ฅฬ = โ
๐ฅ1 + ๐ฅ2 + ๐ฅ๐
๐
๐
๐=1
Tabla Nยฐ 68 Promedio de medidas
K ๐ฅฬ
12.4713
12.5003
12.5133
12.5279
12.4816
12.5074
50. Calcular el error absoluto
๐ธ๐๐๐ ๐ = |๐ฅฬ โ ๐ฅ๐|
28. 28
Ing.DiegoProaรฑoMolina
Tabla Nยฐ69 Cรกlculo del Error Absoluto
K ๐
ฬ ๐ธ๐ด๐ต๐
ฬ ฬ ฬ ฬ ฬ ฬ
12.4713 12.5003 0.029
12.5133 12.5003 0.013
12.5279 12.5003 0.0276
12.4816 12.5003 0.0187
12.5074 12.5003 0.0071
51. Calcular error absoluto medio.
๐ธ๐๐๐
ฬ ฬ ฬ ฬ ฬ = โ
๐ธ๐๐๐ 1 + ๐ธ๐๐๐ 2+๐ธ๐๐๐ ๐
๐
๐
๐=1
Tabla Nยฐ70 Promedio del Error Absoluto
EABS(K) EABS
0.029
0.013
0.0276
0.0187
0.3441
0.0190
52. Calcular el error relativo
Tabla Nยฐ71 Error Relativo
k
๐ธ๐ =
๐ธ๐๐๐
ฬ ฬ ฬ ฬ ฬ
๐ฅฬ
๐ธ๐
=
(0.0190)
(12.5003)
๐ธ๐ = 0.0015
53. Calcular el error porcentual
Tabla Nยฐ72 Error promedio
k
๐ฌ% = ๐ฌ๐ โ ๐๐๐%
๐ธ% = 0.0015 โ 100%
๐ธ% = 0.15 %
Ensayo 11: CALCULO DE LA CONSTANTE DE ELASTICIDAD EN TORNO AL RESORTE 3.
VALORES DE MEDICIรN PARA EL CรLCULO DE VOLUMEN
29. 29
Ing.DiegoProaรฑoMolina
Calculo de la k del resorte
๐ =
๐๐ โ ๐
(๐๐ โ ๐๐)
Tabla Nยฐ 73 Calculo constante K
Masa
kg
0.022 kg 0.043 kg 0.64 kg 0.084 kg 0.088 kg
Li m 0.06 0.06 0.06 0.06 0.06
Lf m 0.068 0.0767 0.0833 0.0906 0.092
K N/m 26.9692 26.8599 26.9376 26.9211 26.9692
Cรกlculo del error absoluto:
54. Calcular la media aritmรฉtica
๐ฅฬ = โ
๐ฅ1 + ๐ฅ2 + ๐ฅ๐
๐
๐
๐=1
Tabla Nยฐ 74 Promedio de medidas
K ๐ฅฬ
26.9692
26.9314
26.8599
26.9376
26.9211
26.9692
55. Calcular el error absoluto
๐ธ๐๐๐ ๐ = |๐ฅฬ โ ๐ฅ๐|
Tabla Nยฐ75 Cรกlculo del Error Absoluto
K ๐
ฬ ๐ธ๐ด๐ต๐
ฬ ฬ ฬ ฬ ฬ ฬ
26.9692 26.9314 0.0378
26.8599 26.9314 0.0715
26.9376 26.9314 0.0062
26.9211 26.9314 0.0103
30. 30
Ing.DiegoProaรฑoMolina
26.9692 26.9314 0.0378
56. Calcular error absoluto medio.
๐ธ๐๐๐
ฬ ฬ ฬ ฬ ฬ = โ
๐ธ๐๐๐ 1 + ๐ธ๐๐๐ 2+๐ธ๐๐๐ ๐
๐
๐
๐=1
Tabla Nยฐ76 Promedio del Error Absoluto
EABS(K) EABS
0.0378
0.0715
0.0062
0.0103
0.0378
0.0327
57. Calcular el error relativo
Tabla Nยฐ77 Error Relativo
k
๐ธ๐ =
๐ธ๐๐๐
ฬ ฬ ฬ ฬ ฬ
๐ฅฬ
๐ธ๐
=
(0.0327)
(26.9314)
๐ธ๐ = 0.0012
58. Calcular el error porcentual
Tabla Nยฐ78 Error promedio
k
๐ฌ% = ๐ฌ๐ โ ๐๐๐%
๐ธ% = 0.0012 โ 100%
๐ธ% = 0.12 %
CALCULO DE LA VELOCIDAD DE LA PARTICULA EN EL PRIMER INSTANTE QUE BAJA
POR LA RAMPA.
Se hace uso de la conservaciรณn de la energรญa para determinar la velocidad.
U1=0.4148 N
U2 = 0.4820 N
33. 33
Ing.DiegoProaรฑoMolina
0.5857 = โ
(
๐ฃ2
๐
)
โฒ
โ (
๐ฃ1
๐
)
โฒ
(
(0)
0.092
) โ (
1.1020
0.0095
)
โ67.9412 = โ(
๐ฃ2
0.092
)
โฒ
+ (
๐ฃ1
0.0095
)
โฒ
โ67.9412 = โ(10.8695 ๐ข2)โฒ + (105.2631 ๐ข1)โฒ
๐๐โฒ =
โ67.9412 + 10.8695 (0.0204)โฒ
๐๐๐.๐๐๐๐
๐๐โฒ = โ ๐.๐๐๐๐ ๐/๐
K1 โ K2 = 12.5003 N
K3 = 26.9314 N
U3 = 1.8314 N
TIEMPO DE BAJADA DEL LA BARRA POR LA RAMPA HASTA LLEGAR AL RESORTE
Mediciรณn de variables para el cรกlculo del tiempo por el que cae la barra hasta llegar al resorte.
Tabla Nยฐ80 tiempo
Nยบ de
ejecuciones
Desplazamiento
X(m)
Tiempo
(s)
1 0.115 0.239
2 0.115 0.239
3 0.115 0.239
4 0.115 0.238
5 0.115 0.239
6 0.115 0.239
7 0.115 0.237
8 0.115 0.239
9 0.115 0.239
10 0.115 0.237
VALORES DE MEDICIรN PARA EL CรLCULO DEL TIEMPO DE BAJADA DE LA BARRA POR
EL PLANO HORIZONTAL 2 HASTA LLEGAR AL RESORTE.
Cรกlculo del error absoluto:
59. Calcular la media aritmรฉtica
๐ฅฬ = โ
๐ฅ1 + ๐ฅ2 + ๐ฅ๐
๐
๐
๐=1
Tabla Nยฐ 81 Promedio de medidas
Tiempo ๐ฅฬ
0.239 0.239 0.2385
35. 35
Ing.DiegoProaรฑoMolina
63. Calcular el error porcentual
Tabla Nยฐ85 Error promedio
Tiempo
๐ฌ% = ๐ฌ๐ โ ๐๐๐%
๐ธ% = 0.0029 โ 100%
๐ธ% = 0.29 %
64. Calculo del rango de valores
Tabla Nยฐ86 Valores mรกx y mรญn
Tiempo
(๐
ฬ ยฑ ๐ฌ๐๐๐
ฬ ฬ ฬ ฬ ฬ )
๐๐๐๐ = (๐. ๐๐๐๐ โ ๐. ๐๐๐๐))
๐๐๐๐ = ๐. ๐๐๐๐
๐๐รก๐ฅ = (๐. ๐๐๐๐ + ๐.๐๐๐๐))
๐๐รก๐ฅ = 0.2392
Tabla Nยฐ 87: Validez de datos
Tiempo Valor mรญnimo Valor mรกximo Valores aceptables
0.239 0.2378 0.2392 Aceptado
0.239 0.2378 0.2392 Aceptado
0.239 0.2378 0.2392 Aceptado
0.238 0.2378 0.2392 Aceptado
0.239 0.2378 0.2392 Aceptado
0.239 0.2378 0.2392 Aceptado
0.237 0.2378 0.2392 Rechazado
0.239 0.2378 0.2392 Aceptado
0.239 0.2378 0.2392 Aceptado
0.237 0.2378 0.2392 Rechazado
ACELERACION DE LA BARRA HASTA LLEGAR AL RESORTE
Mediciรณn de variables para la aceleraciรณn por el que cae la barra hasta llegar al resorte.
๐ =
2โ๐ฅ
๐ก2
Tabla Nยฐ88 Datos del volumen
36. 36
Ing.DiegoProaรฑoMolina
Nยบ de
ejecuciones
Desplazamiento
X(m)
Tiempo
(s)
Aceleraciรณn
(m/s*s)
1 0.115 0.239 4.0265
2 0.115 0.239 4.0265
3 0.115 0.239 4.0265
4 0.115 0.238 4.0604
5 0.115 0.239 4.0265
6 0.115 0.239 4.0265
7 0.115 0.237 4.0947
8 0.115 0.239 4.0265
9 0.115 0.239 4.0265
10 0.115 0.237 4.0947
VALORES DE MEDICIรN PARA EL CรLCULO
Cรกlculo del error absoluto:
65. Calcular la media aritmรฉtica
๐ฅฬ = โ
๐ฅ1 + ๐ฅ2 + ๐ฅ๐
๐
๐
๐=1
Tabla Nยฐ 89 Promedio de medidas
Aceleraciรณn ๐ฅฬ
4.0265 4.0265 4.0435
4.0265 4.0947
4.0265 4.0265
4.0604 4.0265
4.0265 4.0947
66. Calcular el error absoluto
๐ธ๐๐๐ ๐ = |๐ฅฬ โ ๐ฅ๐|
Tabla Nยฐ90 Cรกlculo del Error Absoluto
Aceleraciรณn ๐ฅฬ ๐ธ๐๐๐ ๐
4.0265 0.4044 0.017
4.0265 0.4044 0.017
4.0265 0.4044 0.017
4.0604 0.4044 0.0169
4.0265 0.4044 0.017
4.0265 0.4044 0.017
4.0947 0.4044 0.0512
40. 40
Ing.DiegoProaรฑoMolina
74. Calcular el error relativo
Tabla Nยฐ100 Error Relativo
u
๐ธ๐ =
๐ธ๐๐๐
ฬ ฬ ฬ ฬ ฬ
๐ฅฬ
๐ธ๐ =
(0.0082)
(1.8314)
๐ธ๐ = 0.0044
75. Calcular el error porcentual
Tabla Nยฐ101 Error promedio
u
๐ฌ% = ๐ฌ๐ โ ๐๐๐%
๐ธ% = 0.044 โ 100%
๐ธ% = 0.44 %
76. Calculo del rango de valores
Tabla Nยฐ102 Valores mรกx y mรญn
u
(๐
ฬ ยฑ ๐ฌ๐๐๐
ฬ ฬ ฬ ฬ ฬ )
๐๐๐๐ = (๐. ๐๐๐๐ โ ๐. ๐๐๐๐๐))
๐๐๐๐ = ๐. ๐๐๐๐
๐๐รก๐ฅ = (๐. ๐๐๐๐
+ ๐. ๐๐๐๐๐))
๐๐รก๐ฅ = 1.8396
Tabla Nยฐ 103: Validez de datos
Aceleraciรณn
Valor
mรญnimo
Valor
mรกximo
Valores
aceptables
1.8373 1.8231 1.8396 Aceptado
1.8373 1.8231 1.8396 Aceptado
1.8373 1.8231 1.8396 Aceptado
1.8256 1.8231 1.8396 Aceptado
1.8373 1.8231 1.8396 Aceptado
1.8373 1.8231 1.8396 Aceptado
1.8138 1.8231 1.8396 Rechazado
1.8373 1.8231 1.8396 Aceptado
41. 41
Ing.DiegoProaรฑoMolina
1.8373 1.8231 1.8396 Aceptado
1.8138 1.8231 1.8396 Rechazado
TIEMPO DE DEFORMACION DEL RESORTE CON RESPECTO ALA BARRA QUE CAE.
Mediciรณn de variables para el cรกlculo del tiempo para la deformaciรณn del resorte
Tabla 104 tiempo
Nยบ de
ejecuciones
Tiempo
(s)
1 0.157
2 0.158
3 0.159
4 0.157
5 0.157
6 0.159
7 0.159
8 0.158
9 0.158
10 0.158
VALORES DE MEDICIรN PARA EL CรLCULO DE LA ACELERACION DE LA ESFERA
Cรกlculo del error absoluto:
77. Calcular la media aritmรฉtica
๐ฅฬ = โ
๐ฅ1 + ๐ฅ2 + ๐ฅ๐
๐
๐
๐=1
Tabla Nยฐ 105 Promedio de medidas
Tiempo ๐ฅฬ
0.157 0.159 0.158
0.158 0.159
0.159 0.158
0.157 0.158
0.157 0.158
78. Calcular el error absoluto
๐ธ๐๐๐ ๐ = |๐ฅฬ โ ๐ฅ๐|
Tabla Nยฐ106 Cรกlculo del Error Absoluto
43. 43
Ing.DiegoProaรฑoMolina
82. Calculo del rango de valores
Tabla Nยฐ110 Valores mรกx y mรญn
Tiempo
(๐
ฬ ยฑ ๐ฌ๐๐๐
ฬ ฬ ฬ ฬ ฬ )
๐๐๐๐ = (๐. ๐๐๐ โ ๐. ๐๐๐๐))
๐๐๐๐ = ๐. ๐๐๐
๐๐รก๐ฅ = (๐. ๐๐๐ + ๐.๐๐๐๐))
๐๐รก๐ฅ = 0.1586
Tabla Nยฐ 111: Validez de datos
Tiempo Valor mรญnimo Valor mรกximo Valores aceptables
0.157 0.157 0.1586 Aceptado
0.158 0.157 0.1586 Aceptado
0.159 0.157 0.1586 Rechazado
0.157 0.157 0.1586 Aceptado
0.157 0.157 0.1586 Aceptado
0.159 0.157 0.1586 Rechazado
0.159 0.157 0.1586 Rechazado
0.158 0.157 0.1586 Aceptado
0.158 0.157 0.1586 Aceptado
0.158 0.157 0.1586 Aceptado
G
ANรLISIS DE RESULTADOS
En la realizaciรณn de las tablas iniciamos con la bรบsqueda del tiempo promedio para la esfera que
cae por el Angulo de 34 grados
Tabla Nยฐ 112: Validez de datos
Nยบ de
ejecuciones
Desplazamiento
X(m)
Tiempo
(s)
1 0.3010 0.53
2 0.3010 0.53
3 0.3010 0.54
4 0.3010 0.54
5 0.3010 0.54
44. 44
Ing.DiegoProaรฑoMolina
6 0.3010 0.53
7 0.3010 0.53
8 0.3010 0.53
9 0.3010 0.54
10 0.3010 0.53
De donde obtenemos un valor promedio de
Tabla Nยฐ 113: Validez de datos
Tiempo ๐ฅฬ
0.53 0.53 0.534
0.53 0.53
0.54 0.53
0.54 0.54
0.54 0.53
Del cual se obtuvo un valor para el error porcentual de 0.89 % del cual se analizรณ que es vรกlido ya que
estรก dentro del erro porcentual el cual es el 5%
CALCULO DEL TIEMPO PARA LA ESFERA DESPUES DE BAJAR POR EL PLANO INCLINADO
HASTA LLEGAR A LA BARRA Y COLISIONAR
Obtenemos la siguiente tabla de la cual se toma en cuenta tambiรฉn al desplazamiento para conoc er
cuรกnto fue el desplazamiento de la partรญcula
Tabla Nยฐ 114: Validez de datos
Nยบ de
ejecuciones
Desplazamiento
X(m)
Tiempo
(s)
1 0.1140 0.21
2 0.1140 0.22
3 0.1140 0.21
4 0.1140 0.22
5 0.1140 0.21
6 0.1140 0.21
7 0.1140 0.21
8 0.1140 0.21
9 0.1140 0.21
10 0.1140 0.21
Del cual determinamos un valor promedio para el tiempo en el cual la esfera recorre toda esa distancia y
llega a colisionar con la barra.
Tabla Nยฐ 115: Validez de datos
Tiempo ๐ฅฬ
0.21 0.21 0.212
0.22 0.21
0.21 0.21
45. 45
Ing.DiegoProaรฑoMolina
0.22 0.21
0.21 0.21
Despuรฉs de haber calculado todos los valores obtenemos un error porcentual de 1.5 que de igual manera
estรก dentro del rango del 5% por lo que podemos tomar como vรกlido y podemos usar el valor promedio
para nuestros cรกlculos.
CALCULO DEL TIEMPO DE LA BARRA HASTA LLEGAR A CAER POR LA RAMPA.
Para el cรกlculo del mismo primero se considera que este tiempo ya antes del cรกlculo iba a ser mayor que
el de la esfera y de este encontramos los siguientes datos del tiempo y la distancia que recorrerรญa la barra.
Tabla Nยฐ 116: Validez de datos
Nยบ de
ejecuciones
Desplazamiento
X(m)
Tiempo
(s)
1 0.1200 0.99
2 0.1200 1.00
3 0.1200 1.00
4 0.1200 0.99
5 0.1200 1.00
6 0.1200 0.99
7 0.1200 1.01
8 0.1200 0.99
9 0.1200 1.00
10 0.1200 1.00
Despuรฉs de obtener estos valores se logra obtener un valor promedio del cual obtenemos la siguiente
tabla.
Tabla Nยฐ 117: Validez de datos
Aceleraciรณn ๐ฅฬ
0.99 0.99 0.997
1.00 1.01
1.00 0.99
0.99 1.00
1.00 1.00
Del cual obtenemos un valor porcentual de 0.56 % y damos por hecho que podemos tomar el valor ya
que este estรก dentro del rango de error del 5%.
CALCULAMOS EL VALOR PARA LA ACELERACION DE LA ESFERA QUE BAJA POR EL PLANO
INCLINADO CON UN ANGULO DE 34 GRADOS.
Iniciamos tomando los valores y aรฑadiendo todos los valores como lo son la distancia que recorrerรก asi
tambiรฉn el tiempo de cada uno y determinamos gracias a la siguiente ecuaciรณn para determinar la
aceleraciรณn.
46. 46
Ing.DiegoProaรฑoMolina
Tabla Nยฐ 118: Validez de datos
๐ =
2โ๐ฅ
๐ก2
Nยบ de
ejecuciones
Desplazamiento
X(m)
Tiempo
(s)
Aceleraciรณn
(m/s*s)
1 0.3010 0.53 2.1431
2 0.3010 0.53 2.1431
3 0.3010 0.54 2.0645
4 0.3010 0.54 2.0645
5 0.3010 0.54 2.0645
6 0.3010 0.53 2.1431
7 0.3010 0.53 2.1431
8 0.3010 0.53 2.1431
9 0.3010 0.54 2.0645
10 0.3010 0.53 2.1431
De este determinamos un valor promedio el cual se muestra en la siguiente tabla con ayuda de este valor
se procediรณ a determinar los siguiente valores.
Tabla Nยฐ 119: Validez de datos
Largo ๐ฅฬ
2.1431 2.1431 2.1116
2.1431 2.1431
2.0645 2.1431
2.0645 2.0645
2.0645 2.1431
Y en cuanto al error para determinar si podemos usar el valor fue de 1,78 % y del cual ya podemos hacer
uso el valor promedio ya que el valor mรกximo del error es de 5%.
CALCULO DE LA VELOCIDAD DE LA ESFERA DESPUES DE BAJAR POR EL PLANO INCLINADO
Y LLEGAR A COLISIONAR CON LA BARRA.
Se toma los valores de la distancia y tiempo que ya se tenรญa calculados y los utilizamos para determinar
la aceleraciรณn de la esfera.
Tabla Nยฐ 120: Validez de datos
Nยบ de
ejecuciones
Desplazamiento
X(m)
Tiempo
(s)
Aceleraciรณn
(m/s*s)
1 0.1140 0.21 5.1700
2 0.1140 0.22 4.7107
3 0.1140 0.21 5.1700
4 0.1140 0.22 4.7107
5 0.1140 0.21 5.1700
6 0.1140 0.21 5.1700
7 0.1140 0.21 5.1700
8 0.1140 0.21 5.1700
9 0.1140 0.21 5.1700
47. 47
Ing.DiegoProaรฑoMolina
10 0.1140 0.21 5.1700
Y despuรฉs de ya obtener la aceleraciรณn con la misma fรณrmula con la cual determinamos la aceleraciรณn
que tuvo la misma esfera en el plano inclinado, determine la aceleraciรณn que tendrรญa la esfera cuando
llegue a colisionar con la barra.
Tabla Nยฐ 121: Validez de datos
Aceleraciรณn ๐ฅฬ
5.1700 5.1700 5.0781
4.7107 5.1700
5.1700 5.1700
4.7107 5.1700
5.1700 5.1700
Y ya obtenido el error porcentual procediรณ a calcular los siguientes valores y asรญ obteniendo un valor de
1.5 % para nuestro error porcentual por el cual podemos usar puesto que estรก dentro del 5%..
CALCULO DE LA ACELERACION DE LA BARRA HASTA LLEGAR A CAER POR LA RAMPA 2.
Se toma los valores de la distancia y tiempo y gracias a ellos calculamos la aceleraciรณn gracias a la
fรณrmula que ya hemos utilizado para calcular la aceleraciรณn de la esfera pero para este caso vamos a
calcularlo para la barra y la plasmamos en la siguiente tabla.
Tabla Nยฐ 122: Validez de datos
Nยบ de
ejecuciones
Desplazamiento
X(m)
Tiempo
(s)
Aceleraciรณn
(m/s*s)
1 0.1200 0.99 0.2448
2 0.1200 1.00 0.2400
3 0.1200 1.00 0.2400
4 0.1200 0.99 0.2448
5 0.1200 1.00 0.2400
6 0.1200 0.99 0.2448
7 0.1200 1.01 0.2352
8 0.1200 0.99 0.2448
9 0.1200 1.00 0.2400
10 0.1200 1.00 0.2400
De esta manera se pudo obtener un valor para nuestro valor promedio de la aceleraciรณn y de la cual
podemos hacer uso para la resoluciรณn del ejercicio y encontrar mรกs incรณgnitas que conlleven el uso de la
aceleraciรณn.
Tabla Nยฐ 123: Validez de datos
Aceleraciรณn ๐ฅฬ
0.2448 0.2448 0.24144
0.2400 0.2352
0.2400 0.2448
0.2448 0.2400
0.2400 0.2400
48. 48
Ing.DiegoProaรฑoMolina
Y de los cรกlculos realizado se obtiene un valor del error porcentual del 1.1% el cual estรก dentro del rango
de error y del cual se puede hacer uso ya que el valor mรกximo que no se debe superar el del 5%.
CALCULO DEL COEFICIENTE DE ROZAMIENTO 1
Para el cรกlculo del coeficiente de rozamiento se hizo uno de la siguiente formula despejando las acciones
y reacciones que tenรญa la esfera al momento de bajar por el plano inclinado
๐๐ฅ โ ๐น๐ = ๐ โ ๐
๐ โ ๐๐ ๐๐๐ โ ๐ข โ ๐ = ๐ โ ๐
Tabla Nยฐ124 Datos del volumen
Nยบ de
ejecuciones
Desplazamiento
X(m)
Tiempo
(s)
Aceleraciรณn
(m/s*s)
Peso kg U
1 0.3010 0.53 2.1431 0.018 0.4109
2 0.3010 0.53 2.1431 0.018 0.4109
3 0.3010 0.54 2.0645 0.018 0.4206
4 0.3010 0.54 2.0645 0.018 0.4206
5 0.3010 0.54 2.0645 0.018 0.4206
6 0.3010 0.53 2.1431 0.018 0.4109
7 0.3010 0.53 2.1431 0.018 0.4109
8 0.3010 0.53 2.1431 0.018 0.4109
9 0.3010 0.54 2.0645 0.018 0.4206
10 0.3010 0.53 2.1431 0.018 0.4109
Y obtenidos los coeficientes de rozamiento se procede a calcular el valor promedio del coefiente
de rozamiento que se presenta en la siguiente tabla.
Tabla Nยฐ 125: Validez de datos
u ๐ฅฬ
0.4109 0.4109 0.4148
0.4109 0.4109
0.4206 0.4109
0.4206 0.4206
0.4206 0.4109
En cuanto a este valor promedio se obtuvo un valor del error porcentual del 1.1% el cual esta
dentro del rango mรกximo de 5% que se puede tener de error por lo que podemos usar como nuestro
primer para el coeficiente de rozamiento.
CALCULO DEL COEFICIENTE DE ROZAMIENTO 2
49. 49
Ing.DiegoProaรฑoMolina
Para obtener el valor de este de igual manera se calculรณ con todos los valores que se obtuvieron
para asรญ determinar el coeficiente de restituciรณn que se tendrรญa en torno a la esfera y la madera.
Tabla Nยฐ 126: Validez de datos
Nยบ de
ejecuciones
Desplazamiento
X(m)
Tiempo
(s)
Aceleraciรณn
(m/s*s)
Peso kg U
1 0.1140 0.21 5.1700 0.018 0.4727
2 0.1140 0.22 4.7107 0.018 0.5195
3 0.1140 0.21 5.1700 0.018 0.4727
4 0.1140 0.22 4.7107 0.018 0.5195
5 0.1140 0.21 5.1700 0.018 0.4727
6 0.1140 0.21 5.1700 0.018 0.4727
7 0.1140 0.21 5.1700 0.018 0.4727
8 0.1140 0.21 5.1700 0.018 0.4727
9 0.1140 0.21 5.1700 0.018 0.4727
10 0.1140 0.21 5.1700 0.018 0.4727
Ya obtenido los datos para el coeficiente de rozamiento podemos calcular el valor promedio que
tendrรก para la parte en donde se mueve la esfera.
Tabla Nยฐ 127: Validez de datos
u ๐ฅฬ
0.4727 0.4727 0.4820
0.5195 0.4727
0.4727 0.4727
0.5195 0.4727
0.4727 0.4727
Y despuรฉs de encontrar el valor promedio de u obtuvimos un error porcentual de 3.09% y aunque
este es alto si puede usarse puesto que estรก dentro del error mรกximo del 5%.
CALCULO DE LA CONSTANTE DE ELASTICIDAD PARA EL RESORTE 1 Y 2.
Determine el cรกlculo del resorte 1 y 2 conjuntamente en uno solo puesto que al realizar la toma de las
distancias con los mismos pesos obtuve los valores similares al ser de la misma dimensiรณn los dos, por
lo cual solo se realizรณ un cรกlculo y la constante que se encuentra es tanto para uno como para el otro.
Tabla Nยฐ 127: Validez de datos
Masa
kg
0.022 kg 0.043 kg 0.64 kg 0.084 kg 0.088 kg
Li m 0.080 0.080 0.080 0.080 0.080
Lf m 0.0973 0.1137 0.1301 0.146 0.149
K N/m 12.4713 12.5133 12.5279 12.4816 12.5074
50. 50
Ing.DiegoProaรฑoMolina
Despuรฉs de determinar esos valores podemos determinar el valor de la constante promedio para el
anรกlisis del mismo y obtenemos el siguiente valor.
Tabla Nยฐ 128: Validez de datos
K ๐ฅฬ
12.4713
12.5003
12.5133
12.5279
12.4816
12.5074
Luego de realizar los cรกlculos obtenemos un valor de 12.5003 para el valor promedio de la constante de
elasticidad.
Y para determinar que este valor se lo puede tomar en cuenta determinamos el error porcentual del cual
tuvimos un resultado de 0.15 % que es bajo y estรก dentro del error porcentual del 5% que podemos utilizar
y se considera que podemos usar el valor de la constante de elasticidad tanto para el resorte 1 y 2.
CALCULO DE LA CONSTANTE DE ELASTICIDAD PARA EL Resorte 3
Realizando la obtenciรณn de los valores con diferentes masas obtenemos los siguientes valores los cuales
se procede a buscar un valor promedio.
Tabla Nยฐ 129: Validez de datos
Masa
kg
0.022 kg 0.043 kg 0.64 kg 0.084 kg 0.088 kg
Li m 0.06 0.06 0.06 0.06 0.06
Lf m 0.068 0.0767 0.0833 0.0906 0.092
K N/m 26.9692 26.8599 26.9376 26.9211 26.9692
Determine el valor promedio para una constante elรกstica en torno a el resorte 3 donde el valor que
obtuvimos fue de 26.9314 N y con esto seguimos a encontrar un error porcentual.
Tabla Nยฐ 130: Validez de datos
K ๐ฅฬ
26.9692
26.9314
26.8599
26.9376
26.9211
26.9692
Determinamos un error porcentual de 0.12% el cual sin duda podemos hacer uso de el puesto que este
valor estรก dentro del rango del 5 % que se puede usar, por lo tanto damos por hecho que este valor puede
ser usado.
CALCULO DE LA VELOCIDAD DE LA PARTICULA EN EL PRIMER INSTANTE QUE BAJA POR
LA RAMPA.
51. 51
Ing.DiegoProaรฑoMolina
Para calcular este valor de la velocidad utilizamos la conservaciรณn de la energรญa y como
conocemos la fรณrmula para el cรกlculo de una esfera en dinรกmica rotacional en I cambiamos
por la fรณrmula de
๐
๐
๐๐น๐asรญ para determinar la velocidad cuando baja por la rampa
โ๐ฌ๐ = โ๐ฌ๐ + ๐ธ
๐ฌ๐ = ๐ฌ๐๐น+ ๐ฌ๐๐ป+ ๐ธ๐ + ๐ธ๐ป
๐ โ ๐ โ ๐ =
๐
๐
โ ๐ฐ๐ปโ ๐๐ +
๐
๐
โ ๐ โ ๐๐ + ๐ โ ๐ฐ โ ๐ โ ๐๐๐๐ฝ โ โ๐ฝ + ๐ โ ๐ โ ๐ โ ๐๐๐๐ฝโ โ๐
๐ โ ๐ โ โ =
1
2
โ (
2
5
๐๐ 2 + ๐ โ ๐ 2) โ (
๐ฃ
๐
)
2
+
1
2
โ ๐ โ ๐ฃ2 + ๐ข โ ๐ผ โ ๐ โ ๐๐๐ ๐ โ โ๐ + ๐ข โ ๐ โ ๐ โ ๐๐๐ ๐ โ โ๐
๐ โ โ =
1
2
โ (
7
5
๐ 2) โ (
๐ฃ
๐
)
2
+
1
2
โ ๐ฃ2 + ๐ข โ (
7
5
๐ 2) โ ๐ โ ๐๐๐ ๐ โ
โ๐
๐
+ ๐ข โ ๐ โ ๐๐๐ ๐ โ โ๐
๐ โ โ = (
6
5
๐) โ (๐ฃ)2 + ๐ข โ ๐ โ ๐๐๐ ๐ โ โ๐ (
7
5
โ ๐ + 1)
๐ฃ2 = โ
5
6
(๐ โ โ โ ๐ข โ ๐ โ ๐๐๐ ๐ โ โ๐ (
7
5
) โ ๐ + 1)
๐ฃ = โ
5
6
(9.807 โ 0.18 โ 0.4148 โ 9.807 โ cos(34) โ (
0.18
๐ ๐๐(34)
)(
7
5
) โ (0.0095) + 1)
๐ฃ = 1.5140 ๐/๐
Ahora como llegamos a un plano recto ya no tendremos energรญa potencial sino cinรฉtica y de
igual manera utilizamos la fรณrmula de Inercia del esferas y tenemos una velocidad menor ya
que la partรญcula por el cambio de superficie baja su velocidad al ya estar en un plano recto
๐ฌ๐ = ๐ฌ๐๐น+ ๐ฌ๐๐ป+ ๐ธ๐ + ๐ธ๐ป
๐
๐
๐๐๐ =
๐
๐
โ ๐ฐ๐ป โ ๐๐ +
๐
๐
โ ๐โ ๐๐ + ๐ข โ ๐ผ โ ๐ โ โ๐ + ๐ โ ๐ โ ๐ โ โ๐
๐
๐
๐๐ =
๐
๐
โ (
7
5
๐) โ ๐๐ +
๐
๐
โ ๐๐ + ๐ข โ
7
5
๐ โ ๐ โ โ๐ + ๐ โ ๐ โ โ๐
1
2
โ (
7
5
(0.0095))โ 1.51402
+
1
2
(1.5140 )2
=
1
2
โ (
7
5
(0.0095)) โ v2
+
1
2
โ v2
+ 0.4820 โ
7
5
(0.0095)โ 9.807โ 0.1140+ 0.4820โ 9.807โ 0.1140
๐ = ๐.๐๐๐๐ ๐/๐
52. 52
Ing.DiegoProaรฑoMolina
Velocidad de la barra
Velocidad cero ya que estรก en reposo antes de la colisiรณn
๐ = ๐๐/๐
Velocidad despuรฉs de colisionar, la encontramos gracias a despeje de la velocidad final y las
aceleraciones de donde conocemos todos y lo sustituimos en cada valor, y obtenemos la
velocidad final
โ๐ท = ๐ท๐ โ ๐ท๐ = ๐
(๐ฐ๐+ ๐ฐ๐)โ ๐๐ โ (๐ฐ๐โ ๐๐+ ๐ฐ๐ โ ๐๐) = ๐
(๐ฐ๐+ ๐ฐ๐)โ ๐๐ = (๐ฐ๐ โ ๐๐ + ๐ฐ๐ โ ๐๐)
๐ฝ๐ =
๐ฐ๐โ ๐๐ + ๐ฐ๐ โ ๐๐
๐ฐ๐+ ๐ฐ๐
๐๐ =
(
7
5
(0.018)(0.0095)) โ (๐.๐๐๐๐) + (
๐.๐๐๐โ๐.๐๐๐๐
๐
)(
๐
๐.๐๐๐๐
)
(
7
5
0.018(0.0095)) + (
๐.๐๐๐(๐.๐๐๐)๐
๐
)
๐๐ =
(
7
5
(0.018)(0.0095)) โ (๐.๐๐๐๐) + (
๐.๐๐๐โ๐.๐๐๐๐
๐
)(
๐
๐.๐๐๐๐
)
(
7
5
0.018(0.0095)) + (
๐.๐๐๐(๐.๐๐๐)๐
๐
)
๐๐ = ๐.๐๐๐๐ ๐/๐
Calculo del coeficiente de restituciรณn, como esto sucede antes del choque entonces la
velocidad para el objeto dos no existe entonces se elimina y solo tendrรญamos la velocidad de
u1 y v1 y los dividimos y obtenemos que es un choque elรกstico por el coeficiente de
restituciรณn.
๐ = โ
(๐โฒ๐โ ๐โฒ๐)
(๐๐ โ ๐๐)
๐ =
๐โฒ๐
๐๐
๐ =
๐.๐๐๐๐
๐.๐๐๐๐
53. 53
Ing.DiegoProaรฑoMolina
๐ = ๐.๐๐๐๐
Coeficiente entre 0<e<1
Por lo tanto es un coeficiente semi-elรกstico.
Calculo de las velocidades despuรฉs del impacto de la esfera con la barra. Como ya
obtuvimos el coeficiente de restituciรณn ya podemos intercambiarlo en la fรณrmula de choques
7
5
(0.018)(0.0095) โ (1.1020) +
0.060 โ 0.0922
3
โ (
0
0.0150
)
=
7
5
(0.018)(0.0095) โ ๐ข1 ยด +
0.060 โ 0.0922
3
โ
๐ข2
0.0105
ยด
0.0002 = 0.0002 โ (
โ67.9412 + 10.8695 ๐ข2โฒ
๐๐๐.๐๐๐๐
) + 0.0161 ๐ข2ยด
0.0002 = 0.0002(โ0.6454 + 0.1032 ๐ข2โฒ) + 0.0161 ๐ข2ยด
0.0002 = โ0.0001 + 0.00002 ๐ข2โฒ + 0.0161 ๐ข2ยด
๐๐ = ๐. ๐๐๐๐ ๐/๐
๐ = โ
(
๐ฃ1
๐
)
โฒ
โ (
๐ฃ2
๐
)
โฒ
(
๐ฃ1
๐
) โ (
๐ฃ2
๐
)
0.5857 = โ
(
๐ฃ2
๐
)
โฒ
โ (
๐ฃ1
๐
)
โฒ
(
(0)
0.092
) โ (
1.1020
0.0095
)
โ67.9412 = โ (
๐ฃ2
0.092
)
โฒ
+ (
๐ฃ1
0.0095
)
โฒ
โ67.9412 = โ(10.8695 ๐ข2)โฒ + (105.2631 ๐ข1)โฒ
๐๐โฒ =
โ67.9412 + 10.8695 (0.0204)โฒ
๐๐๐.๐๐๐๐
๐๐โฒ = โ ๐.๐๐๐๐ ๐/๐
Y asรญ determinamos la velocidad que tendrรก los dos objetos despuรฉs de colisionar.
54. 54
Ing.DiegoProaรฑoMolina
DETERMINAR EL TIEMPO EN QUE LA BARRA BAJA HASTA LLEGAR AL TOCAR AL RESORTE.
Primero determinamos el tiempo en que y lo colocamos en la siguiente tabla.
Tabla Nยฐ 131: Validez de datos
Nยบ de
ejecuciones
Desplazamiento
X(m)
Tiempo
(s)
1 0.115 0.239
2 0.115 0.239
3 0.115 0.239
4 0.115 0.238
5 0.115 0.239
6 0.115 0.239
7 0.115 0.237
8 0.115 0.239
9 0.115 0.239
10 0.115 0.237
Y con estos valores encontramos un tiempo promedio en el cual baja la partรญcula el cual es de 0.2385 s y
se procediรณ a calcular el valor del error porcentual para determinar si este valor estaba dentro del rango
del error porcentual el cual fue de 0.29% que nos dice que nuestro error es bajo y estรก por debajo del 5%
por lo cual puede ser usado y es vรกlido.
CALCULO DE LA ACELERACIรN DE LA BARRA DURANTE EL TIEMPO QUE BAJA POR LA RAMPA
2 HASTA LLEGAR A TOCAR AL RESORTE.
Iniciamos con el tiempo que ya obtuvimos y determinamos cual es la aceleraciรณn de cada tiempo
obteniendo la siguiente tabla donde tenemos los valores de la aceleraciรณn.
Tabla Nยฐ 132: Validez de datos
Nยบ de
ejecuciones
Desplazamiento
X(m)
Tiempo
(s)
Aceleraciรณn
(m/s*s)
1 0.115 0.239 4.0265
2 0.115 0.239 4.0265
3 0.115 0.239 4.0265
4 0.115 0.238 4.0604
5 0.115 0.239 4.0265
6 0.115 0.239 4.0265
7 0.115 0.237 4.0947
8 0.115 0.239 4.0265
9 0.115 0.239 4.0265
10 0.115 0.237 4.0947
Despuรฉs de obtener estos valores determinamos el valor promedio de la aceleraciรณn el cual obtuvimos
con el valor de 4.0435 m/s*s para nuestra aceleraciรณn y con esto se procediรณ a buscar el error porcentual
el cual fue de 0.58 % el cual es valor que estรก dentro del rango de valor valido y se lo puede usar.
CALCULO DEL COEFICIENTE DE ROZAMIENTO DE LA BARRA CON RESPECTO AL CARTON.
55. 55
Ing.DiegoProaรฑoMolina
Calculamos los valores de la fuerza de rozamiento con la siguiente formula y en la tabla como ya
obtenemos los valores que necesitamos procedemos a calcular cada uno de ellos y colocarlo en el
coeficiente de rozamiento.
๐๐ฅ โ ๐น๐ = ๐ โ ๐
๐ โ ๐ โ ๐ข โ ๐ = ๐ โ ๐
Tabla Nยฐ133 Datos del volumen
Nยบ de
ejecuciones
Desplazamiento
X(m)
Tiempo
(s)
Aceleraciรณn
(m/s*s)
Peso kg U
1 0.115 0.239 4.0265 0.060 1.8373
2 0.115 0.239 4.0265 0.060 1.8373
3 0.115 0.239 4.0265 0.060 1.8373
4 0.115 0.238 4.0604 0.060 1.8256
5 0.115 0.239 4.0265 0.060 1.8373
6 0.115 0.239 4.0265 0.060 1.8373
7 0.115 0.237 4.0947 0.060 1.8138
8 0.115 0.239 4.0265 0.060 1.8373
9 0.115 0.239 4.0265 0.060 1.8373
10 0.115 0.237 4.0947 0.060 1.8138
Y determinamos el valor promedio del coeficiente de rozamiento el cual obtuvimos que fue de 1.8314 N y
encontramos el error porcentual que fue de aproximadamente de 0.44% el cual es vรกlido ya que estรก
dentro del error porcentual del 5%.
DETERMINAR EL TIEMPO DE COMPRESION DEL RESORTE DESPUES DE QUE LA BARRA HAYA
CAIDO SOBRE EL.
Primero tomamos las medidas de los tiempos y los colocamos en un tabla para conocer cuรกles fueron los
tiempos que este demoro en comprimirse.
Tabla Nยฐ 134: Validez de datos
Nยบ de
ejecuciones
Tiempo
(s)
1 0.157
2 0.158
3 0.159
4 0.157
5 0.157
6 0.159
7 0.159
8 0.158
9 0.158
10 0.158
CALCULAR EL TIEMPO EN EL CUAL EL RESORTE SE COMPRIME CON LA BARRA.
56. 56
Ing.DiegoProaรฑoMolina
Para este paso consideramos que si la barra es devuelta por el resorte, pero para este caso no hubo
dicho efecto sino que la barra se comprimiรณ y no hubo ningรบn movimiento por lo cual solo se tomรณ el
tiempo sin dicho efecto de que la barra se mueva despuรฉs de comprimir todo el resorte y colocamos los
valores en esta tabla.
Tabla Nยฐ 135: Validez de datos
Nยบ de
ejecuciones
Desplazamiento
X(m)
Tiempo
(s)
1 0.3010 0.157
2 0.3010 0.158
3 0.3010 0.159
4 0.3010 0.157
5 0.3010 0.157
6 0.3010 0.159
7 0.3010 0.159
8 0.3010 0.158
9 0.3010 0.158
10 0.3010 0.158
Del cual al realizar los respectivos cรกlculos obtuvimos un tiempo promedio de 0.158 s y ademรกs para
validar este valor se procediรณ a determinar el valor del erros porcentual el cual fue de 0.37 % el cual es
vรกlido ya que estรก dentro del rango del valor porcentual del 5% que se puede utilizar.
PREGUNTAS:
1. ยฟQuรฉ es dinรกmica rotacional?
La dinรกmica rotacional es el estudio del movimiento de rotaciรณn teniendo en cuenta otro movimiento que
estรฉ ocurriendo con el cuerpo rรญgido.
2. ยฟQuรฉ es la velocidad angular?
Se expresa como el รกngulo girado por unidad de tiempo y se mide en radianes por segundo. La rotaciรณn
es una propiedad vectorial de un cuerpo. El vector representativo de la velocidad angular es paralelo a la
direcciรณn del eje de rotaciรณn y su sentido indica el sentido de la rotaciรณn siento el sentido anti horario.
3. ยฟEs necesario encontrar el error porcentual (Si o No) para validar cada uno de los datos que se obtuvo?
Si es necesario ya que asรญ podemos saber si nuestro error estรก dentro del error porcentual mรกximo al cual
debe estar nuestro proyecto.
4. ยฟQuรฉ es inercia rotacional?
Es una medida de la inercia rotacional de un cuerpo. El momento de inercia desempeรฑa en la
rotaciรณn el papel que la masa desempeรฑa en la traslaciรณn.
57. 57
Ing.DiegoProaรฑoMolina
5. ยฟCuรกl es la ecuaciรณn de dinรกmica rotacional para una esfera?
2
5
๐๐ 2
6. Que es el equilibrio rotacional
El concepto de equilibrio rotacional es el equivalente de la primera ley de Newton para un sistema en
rotaciรณn. Un objeto que no estรก girando continua sin rotar a menos que una torca externa actuรฉ sobre รฉl,
del mismo modo un objeto que gira a velocidad angular constante continรบa rotando a menos que una
torca externa actuรฉ sobre รฉl.
7. Como podemos encontrar la inercia de formas complejas.
Para encontrar la inercia rotacional de figuras mรกs complicadas generalmente es necesario usar el
cรกlculo. Sin embargo para muchas formas geomรฉtricas comunes, en libros de texto u otras fuentes, es
posible encontrar tablas con fรณrmulas para inercia rotacional.
8. ยฟCuรกl fue el coeficiente de rozamiento de los tres materiales que se usรณ en la maqueta.?.
๐ข1 = 0.4148 ๐
๐ข2 = 0.4820 ๐
๐ข3 = 1.8314 ๐
9. ยฟCuรกl fue la velocidad antes del impacto de la esfera y de la barra?
๐ฃ1 = 1.1020
๐
๐
๐ฃ2 = 0 ๐/๐
La velocidad de la barra es cero puesto que esta se encontraba en reposo.
10. ยฟCuรกl fue el coeficiente de restituciรณn que se encontrรณ ?.
๐ = ๐.๐๐๐๐
Este coeficiente de restituciรณn nos da a notar que el choque fue semi- elรกstico.
5 CONCLUSIONES
๏ท Con la realizacion de la maqueta de dinamica rotacional se identifico diferentes fenomenos fisicos los
cuales mediante la aplicaciรณn de la teoria de errores se fueron encontrando valores promedio los
cuales utilizamos para determinar valores exactos para cada uno de los sucesos para encontrar el
58. 58
Ing.DiegoProaรฑoMolina
tiempo, los coeficientes de rozamiento asi como el coeficiente de restitucion del choque entres los
objetos y las constantes elasticas del sistema de resortes.
๏ท Gracias al calculo de errores se fue identificando que sucedian asi como lo es el tiempo de bajada de
la esfera y de tiempo antes del choque de la esfera con la barra y el tiempo en cual la barra fue hasta
caer a la segunda rampa y comprimir el sistema de resortes.y los tiempor promedio que obtuvimos
fuero de ๐ก1 = 0.534 ๐ ; ๐ก2 = 0.212 ๐ ;๐ก3 = 0.997 ๐ ; ๐ก4 = 0.2385 ๐ ;๐ก5 = 0.158 ๐ de esta manera
podemos conocer los tiempos promedios en los cuales sucediรณ cada uno de procesos tanto de la
esfera como de la barra y de los resortes.
๏ท Como utilizamos tres tipos de materiales esperabamos que estos variaran ademas del uso de los
materiales de los objetos que utilizamos para esta simulacion, y ya determinados los tiempos y
aceleraciones con ayuda de la dinamica pudimos utilizar la formula de โ ๐ฆ = 0 ๐ฆ โ ๐ฅ = ๐๐ con la
cual determinamos con los datos de los pesos y aceleraciones que ya conociamos la fuerza de
rozamiento que fue para el cristal de ๐ข1 = 0.4148 ๐ para la madera de 0.4820 ๐ y por ultimo el
carton de 1.8314 ๐.
๏ท En cuanto al valor del coeficiente de rozamiento encontramos un valor de ๐ = 0.5857 por el cual ya
podemos determinar que si 0 < ๐ < 1 es un coeficiente de restitucion semi elastico.
๏ท Para todos los calculos se hizo uso del error maximo del 5% y el unico valor que fue de mayor grado
de error pero no sobrepasaba el valor del error porcentual fue el de aproximadamente 3% que si esta
dentro del error pocentual lo consideramos valido.
6 RECOMENDACIONES
๏ท Utilizar un materiales los cuales sean faciles de controlar para poder construir la maqueta mas
facilmente ademas de pedir ayuda a otra persona para colocar cada una de las piezas de la
maqueta.
๏ท Para la toma del tiempo utilizar un cronometro el cual sea efectivo y no un celular ya que este no
capta mas decimales ademas de igual manera pedir ayuda a una persona e ir tomando los tiempos
por tramos.
๏ท Para encontrar los coeficientes de rozamiento tomar en cuenta la aceleracion la cual debemos
obtener con el tiempo.
๏ท Si en caso de que el coeficiente de restituciรณn sobrepase 1 o sea menor que cero considerar la
revisiรณn de cada uno de los datos.
๏ท Utilizar materiales los cuales sean medianamente exactos para obtener valores menores y que no
sobrepasen el 5%
7 REFERENCIAS BIBLIOGRรFICAS Y DE LA WEB
1. Diego-profesor , โDinรกmica rotacional, Segunda ley de Newton para la rotaciรณn ejemplo 1โ ,
https://www.youtube.com/watch?v=qNa8g3j3PBs consultado19 03 2021.
2. Edison Naranjo, โDinรกmica Rotacionalโ , https://www.slideshare.net/EdisonNaranjo3/dinamica-
rotacional-5066938, consultado 19 03 2021.
59. 59
Ing.DiegoProaรฑoMolina
3. Sbweb. โ Ecuaciรณn de la dinรกmica de rotaciรณnโ,
http://www.sc.ehu.es/sbweb/fisica/solido/teoria/teoria.htm, Consultado 19 03 2021
4. ESPOL, โDinรกmica de la rotaciรณnโ, https://www.dspace.espol.edu.ec/bitstream/1234
56789/6615/1/Din%C3%A1mica%20de%20la%20rotaci%C3%B3n.pdf Consultado 19 03 2021
5. PUCC, โDinรกmica de rotaciones โ, http://fisica.uc.cl/images/Exp_5-Dinamica_de_Rotaciones.pdf ,
Consultado 19 03 2021.
6. N. Beraha, M.F.Carusela y C.D.El Has, โDinรกmica del movimiento rotacional: propuesta de experiencias
sencillas para facilitar su comprensiรณnโ, https://www.scielo.br/pdf/rbef/v31n4/v31n4a17.pdf , consultado
19 03 2021.
7. Wiki Libros , โFรญsica / dinรกmica de rotaciรณnโ,
https://es.wikibooks.org/wiki/F%C3%ADsica/Din%C3%A1mica_de_rotaci%C3%B3n., Consultado 19
03 2021.
8. M.L Peรฑafiel (2015), โDinรกmica rotacional relativistaโ ,
http://www.scielo.org.bo/scielo.php?script=sci_arttext&pid=S1562-38232015000100001, Consultado
19 03 2021.
9. J Mena, โDinรกmica Rotacionalโ , https://www.monografias.com/trabajos107/dinamica-
rotacional/dinamica-rotacional.shtml, Consultado 19 03 2021
10. FisicaLab, Segunda ley de Newton Aplicada a la Rotaciรณn de un Solido ,
https://www.fisicalab.com/apartado/segunda-ley-newton-rotacion, Consultado 19 03 2021.
11. Wikipedia โMovimiento Rotatorioโ, https://es.wikipedia.org/wiki/Movimiento_rotativo Consultado 19 03
2021.
12. Khan Academy , โInercia rotacionalโ, https://es.khanacademy.org/science/physics/torque-angular-
momentum/torque-tutorial/a/rotational-inertia, Consultado 19 03 2021.
13. UNIVERSIDAD NACIONAL EXPERIMENTAL POLITรCNICA DE LA FUERZA ARMADA
BOLIVARIANA , Guia practica de Fรญsica I, http://electricamaracay.webcindario.com/pulido/guia5.1.pdf,
Consultado 19 03 2021.
14. Elvis Mendoza, โDinรกmica Rotacionalโ, https://sites.google.com/site/dinamicadelarotacion/unidad-i-
introduccion-a-el-lenguaje-c, Consultado 19 03 2021.
15. M Navarro , โDinรกmica de un sรณlido rรญgidoโ,
https://w3.ual.es/~mnavarro/Tema3Dinamicasolidorigido.pdf, Consultado 19 03 2021.
16. XMind , Dinรกmica Rotacional, https://www.xmind.net/embed/GPTK/#, Consultado 19 03 2021.
17. T David Navarrete Gonzales (1996) (Colecciรณn de problemas resueltos para el curso de dinรกmica ),
https://core.ac.uk/reader/48392280, Consultado 19 03 2021.
18. Juan Felipe M, Resumen Dinรกmica Rotacional , https://es.scribd.com/doc/99442277/RESUMEN-
DINAMICA-ROTACIONAL , Consultado 19 03 2021.
19. Dr. Roberto Pedro Duarte Zamorano (2020), โMecรกnica IIโ,
https://rpduarte.fisica.uson.mx/archivos/curso1/02-MecanicaII.pdf , Consultado 19 03 2021.
20. Javier Junquera, Movimiento de Rotaciรณn,
https://personales.unican.es/junqueraj/javierjunquera_files/fisica-1/10.movimiento-de-rotacion.pdf,
Consultado 19 03 2021.