1. The document discusses work, energy, and their relationship as described by the work-energy theorem. It defines work as a force applied over a displacement, and gives the equation W=Fd.
2. Kinetic energy is defined as K=1/2mv^2. The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy, or W=ΔK.
3. Potential energy, such as gravitational potential energy mgh, is discussed. The work done by gravity in lifting an object does not depend on the path, only the change in height.
The document contains multiple conceptual physics problems involving conservation of energy.
1) A system of two cylinders connected by a cord over a frictionless peg is released from rest. The system's mechanical energy is conserved, so the potential energy decreases and kinetic energy increases.
2) Estimation problems calculate the minimum time to climb stairs or the Empire State Building using assumptions about maximum metabolic rate and efficiency.
3) Multiple other problems apply conservation of mechanical energy to calculate quantities like tension, equilibrium angles, or speeds in various physical systems like swings or circular motion.
- A block rests on an inclined plane making an angle of 30° with the horizontal.
- Static friction acts on the block to prevent it from sliding down the plane.
- The coefficient of static friction (μs) between the block and plane can be expressed as μs = tanθ.
- Since the plane makes an angle of 30° with the horizontal, the coefficient of static friction is equal to tan30° = 0.577.
The document discusses work, kinetic energy, and power as they relate to mechanical systems. It includes conceptual problems with explanations of the relevant physics concepts and mathematical solutions. Specifically:
- Problem 7 compares the work required to stretch a spring different distances based on the equation that work done on a spring is proportional to the square of its displacement.
- Problem 13 tests understanding of scalar products by asking whether several statements about scalar products and vectors are true or false.
- Problem 17 explains that the only external force doing work to accelerate a car from rest is friction between the tires and the road, using free body diagrams and the work-kinetic energy theorem.
Principles of soil dynamics 3rd edition das solutions manualHuman2379
Full download: https://goo.gl/MyzREj
principles of soil dynamics pdf
soil dynamics and liquefaction
fundamentals of soil dynamics and earthquake engineering
principles of foundation engineering
The document discusses work, energy, and the work-energy principle as an alternative way to analyze motion compared to using forces and Newton's laws. It defines key terms like work, kinetic energy, and systems. The work-energy principle states that the net work done on an object equals its change in kinetic energy (Wnet = ΔKE). This allows reexpressing Newton's second law in terms of energy rather than forces. Examples show how to calculate work, kinetic energy, and use the work-energy principle to solve motion problems.
Work is the transfer of energy by a force acting on an object. Work done is calculated as force times distance moved in the direction of the force. Kinetic energy is the energy of an object due to its motion, calculated as one-half mass times velocity squared. Potential energy is energy due to an object's position or state, such as gravitational potential energy which depends on mass and height. The total energy in a system remains constant according to the law of conservation of energy, though the form of energy may change. Power is the rate at which work is done or energy is transferred, calculated as work divided by time.
This document discusses Castigliano's theorems for analyzing stresses and strains in structures. It explains that Castigliano's first theorem states that the partial derivative of a structure's strain energy with respect to an applied force equals the displacement at the point of application of that force. Castigliano's second theorem states that the partial derivative of strain energy with respect to a displacement equals the force that produces that displacement. The document provides mathematical expressions to calculate strain energy and uses these theorems to analyze beam deflections under applied loads.
The document provides conceptual problems and their solutions related to Newton's Laws of motion.
1) A problem asks how to determine if a limousine is changing speed or direction using a small object on a string. The solution is that if the string remains vertical, the reference frame is inertial.
2) Another problem asks for two situations where apparent weight in an elevator is greater than true weight. The solution states this occurs when the elevator accelerates upward, either slowing down or speeding up.
3) A third problem involves forces between blocks and identifies which constitute Newton's third law pairs. The normal forces between blocks and between a block and table are identified as third law pairs.
The document contains multiple conceptual physics problems involving conservation of energy.
1) A system of two cylinders connected by a cord over a frictionless peg is released from rest. The system's mechanical energy is conserved, so the potential energy decreases and kinetic energy increases.
2) Estimation problems calculate the minimum time to climb stairs or the Empire State Building using assumptions about maximum metabolic rate and efficiency.
3) Multiple other problems apply conservation of mechanical energy to calculate quantities like tension, equilibrium angles, or speeds in various physical systems like swings or circular motion.
- A block rests on an inclined plane making an angle of 30° with the horizontal.
- Static friction acts on the block to prevent it from sliding down the plane.
- The coefficient of static friction (μs) between the block and plane can be expressed as μs = tanθ.
- Since the plane makes an angle of 30° with the horizontal, the coefficient of static friction is equal to tan30° = 0.577.
The document discusses work, kinetic energy, and power as they relate to mechanical systems. It includes conceptual problems with explanations of the relevant physics concepts and mathematical solutions. Specifically:
- Problem 7 compares the work required to stretch a spring different distances based on the equation that work done on a spring is proportional to the square of its displacement.
- Problem 13 tests understanding of scalar products by asking whether several statements about scalar products and vectors are true or false.
- Problem 17 explains that the only external force doing work to accelerate a car from rest is friction between the tires and the road, using free body diagrams and the work-kinetic energy theorem.
Principles of soil dynamics 3rd edition das solutions manualHuman2379
Full download: https://goo.gl/MyzREj
principles of soil dynamics pdf
soil dynamics and liquefaction
fundamentals of soil dynamics and earthquake engineering
principles of foundation engineering
The document discusses work, energy, and the work-energy principle as an alternative way to analyze motion compared to using forces and Newton's laws. It defines key terms like work, kinetic energy, and systems. The work-energy principle states that the net work done on an object equals its change in kinetic energy (Wnet = ΔKE). This allows reexpressing Newton's second law in terms of energy rather than forces. Examples show how to calculate work, kinetic energy, and use the work-energy principle to solve motion problems.
Work is the transfer of energy by a force acting on an object. Work done is calculated as force times distance moved in the direction of the force. Kinetic energy is the energy of an object due to its motion, calculated as one-half mass times velocity squared. Potential energy is energy due to an object's position or state, such as gravitational potential energy which depends on mass and height. The total energy in a system remains constant according to the law of conservation of energy, though the form of energy may change. Power is the rate at which work is done or energy is transferred, calculated as work divided by time.
This document discusses Castigliano's theorems for analyzing stresses and strains in structures. It explains that Castigliano's first theorem states that the partial derivative of a structure's strain energy with respect to an applied force equals the displacement at the point of application of that force. Castigliano's second theorem states that the partial derivative of strain energy with respect to a displacement equals the force that produces that displacement. The document provides mathematical expressions to calculate strain energy and uses these theorems to analyze beam deflections under applied loads.
The document provides conceptual problems and their solutions related to Newton's Laws of motion.
1) A problem asks how to determine if a limousine is changing speed or direction using a small object on a string. The solution is that if the string remains vertical, the reference frame is inertial.
2) Another problem asks for two situations where apparent weight in an elevator is greater than true weight. The solution states this occurs when the elevator accelerates upward, either slowing down or speeding up.
3) A third problem involves forces between blocks and identifies which constitute Newton's third law pairs. The normal forces between blocks and between a block and table are identified as third law pairs.
This document discusses simple harmonic motion and elasticity. It covers topics like Hooke's law, springs, oscillations, energy related to springs, pendulums, damping, driven harmonic motion and resonance. Several examples are provided to illustrate concepts like calculating restoring forces on springs, determining natural frequencies, and relating stress and strain using Hooke's law.
1) The document discusses the principle of virtual work which is used to calculate deflections in statically determinate and indeterminate structures.
2) It defines virtual work and differentiates between external and internal virtual work. The principle of virtual displacement and virtual forces are also introduced.
3) The unit load method for calculating deflections is described. This involves considering a structure under a system of actual loads that produce real displacements, and a separate system of virtual loads that produce virtual displacements.
1) A student pointed out that a sailboat with a fan blowing into its sails could in fact move forward, because the air molecules bouncing off the sail would impart twice the momentum change experienced going through the fan, providing a net forward force.
2) A 2000-kg car traveling at 30 m/s collided perfectly inelastically with another 2000-kg car traveling at 10 m/s, sticking together with a final speed of 20 m/s. 20% of the initial kinetic energy was lost to heat and deformation.
3) A 16-g bullet fired into a 1.5-kg ballistic pendulum was calculated to have a speed of 45 m/s before impact by applying
1) The document discusses the work-energy principle, which states that the work done on an object equals its change in kinetic energy.
2) It provides various examples of calculating work done by different types of forces like inclined, gravitational, frictional, and spring forces.
3) Key concepts covered include the definitions of work and work done, sign conventions for different forces, and using the work-energy principle to solve problems by setting the work equal to the change in kinetic energy.
Principles of soil dynamics 3rd edition das solutions manualTuckerbly
Download at: https://goo.gl/NfK7oK
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principles of soil dynamics 3rd edition pdf
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principle of soil dynamics
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principles of foundation engineering
The document discusses work, energy, and the conservation of mechanical energy. It defines work as the product of force and displacement, and introduces kinetic energy as the energy of motion and potential energy as stored energy due to position or force interactions. The document also explains that the total mechanical energy, which is the sum of an object's kinetic and potential energies, remains constant in an isolated system according to the law of conservation of energy.
Work is defined as a force causing displacement. James Joule discovered the relationship between work, energy, and temperature. Mechanical energy exists in kinetic and potential forms. Kinetic energy depends on an object's motion while potential energy depends on its position or elastic source. Work and energy calculations can be used to determine values like spring constant and power.
The document provides an introduction to work, energy and power in physics. It defines work as the product of the component of a force in the direction of displacement and the magnitude of displacement. Kinetic energy is defined as half the mass times the square of an object's speed. The work-energy theorem states that the change in an object's kinetic energy is equal to the work done on it by net force. Positive work is done when force and displacement are in the same direction, and negative work is done when they are in opposite directions. Power is defined as the rate of doing work or energy transfer.
This document discusses work, energy, and the conservation of mechanical energy. It defines work as the product of force and displacement, with units of joules. Kinetic energy depends on an object's mass and speed, and potential energy includes gravitational potential energy and elastic potential energy. The principle of conservation of mechanical energy states that the total mechanical energy of an isolated system remains constant over time. Examples are provided to demonstrate calculations of work, kinetic energy, potential energy, and applying the conservation of mechanical energy to solve for unknown values.
This document discusses key concepts of work, energy, and power. It defines energy as the capacity to do work and defines work in physics as the application of a force through a distance. Units of energy are joules, which equal kg-m^2/s^2. Kinetic energy is defined as 1/2mv^2 and represents the energy of motion. Gravitational potential energy is defined as mgh and represents energy stored by raising a mass against gravity. Examples are provided to illustrate kinetic energy, gravitational potential energy, and how energy is transferred and transformed between the two.
Strong Nuclear Force and Quantum Vacuum as Gravity (FUNDAMENTAL TENSOR)SergioPrezFelipe
Publication at ccsenet. Gravity explained by a new theory, ‘Superconducting String Theory (SST)’, completely opposite from current field emission based and inspired on originals string theories. Strengths are decomposed to make strings behave as one-dimensional structure with universe acting as a superconductor where resistance is near 0 and the matter moves inside. Strong nuclear force, with an attraction of 10.000 Newtons is which makes space to curve, generating acceleration, more matter more acceleration. Electromagnetic moves in 8 decimals, gravity is moved to more than 30 decimals to work as a superconductor.
- Work is done when a force causes an object to move in the direction of the force. No work is done if there is no movement.
- Work (W) is calculated as the product of the magnitude of the force (F) and the magnitude of displacement (d) in the direction of the force.
- Power is the rate at which work is done and is calculated as work (W) divided by time (t). It is measured in watts, with 1 watt equaling 1 joule per second.
The document discusses Newton's second law of motion which states that acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, and inversely proportional to the mass of the object. It provides examples of calculating the acceleration of objects given their mass and applied net force using the equation a=Fnet/m. These examples include kicking a football, lifting an elevator car with passengers, and forces acting on an object from different directions.
The document discusses impulse, momentum, and collisions. It defines impulse as the product of an average force and the time it acts, and momentum as the product of an object's mass and velocity. The impulse-momentum theorem states that impulse equals change in momentum when a net force acts. Conservation of momentum means the total momentum of an isolated system remains constant. Collisions can be elastic or inelastic, depending on whether kinetic energy is conserved.
Work, energy, and power are defined. Work is force times distance. Kinetic energy is equal to one-half mass times velocity squared. Power is the rate of doing work, defined as work per unit time. Several examples of calculating work are provided for different scenarios like lifting an object, springs, and overcoming friction. The key concepts of work, kinetic energy, and power are reviewed.
1) The document provides calculations for force (894 N), acceleration (9.93 m/s^2), kinetic energy (50J), velocity (10 m/s), momentum, work (22782.06 J), and power using various physics equations.
2) It identifies that the pairs of interacting objects in different scenarios involving a rocket are the rocket and exhaust gases, exhaust gases and air, and gravity, rocket exhaust, and friction.
3) It states that compression waves are most clearly seen in longitudinal waves like a slinky where the coils are close together.
The document discusses mechanical energy and its two forms: potential and kinetic energy. Potential energy is energy due to an object's position or state, like a raised weight or stretched spring, while kinetic energy is energy due to an object's motion. Mechanical energy, the energy due to an object's position or motion, is the sum of its potential and kinetic energies. Examples are given like a bow's potential energy being transferred to an arrow's kinetic energy on release. The relationships between work, potential and kinetic energy are also explained.
1. Work done by a constant force depends on the magnitude of the force and the displacement along the direction of the force. Work done by opposing forces is negative. Centripetal forces do no work as they are always perpendicular to motion.
2. The work-kinetic energy theorem states that the work done on an object equals its change in kinetic energy. It can be used to calculate changes in speed.
3. Gravitational potential energy is defined as mgh. The principle of conservation of mechanical energy can be used to solve problems involving changes in kinetic and potential energy in an isolated system, such as an object moving under gravity.
With this mantra success is sure to come your way. At APEX INSTITUTE we strive our best to realize the Alchemist's dream of turning 'base metal' into 'gold'.
B conservative and non conservative forcesdukies_2000
This document discusses conservative and non-conservative forces, and the principles of conservation of energy and mechanical energy. It states that for conservative forces, the total energy within a closed system remains the same, though it can transform between potential and kinetic forms. For conservative forces, the net work over a closed loop is zero, and the work is path independent. Friction is a non-conservative force where net work is done over a closed loop and more work is done over longer distances. Potential energy is the other form of energy involved in conservative systems, where the sum of potential and kinetic energy equals the total energy and changes in one form equal negative changes in the other.
Karen Adelan presented on the topic of classical mechanics and energy. Some key points:
- Energy is a conserved quantity that can change forms but is never created or destroyed. It is useful for describing motion when Newton's laws are difficult to apply.
- Kinetic energy is the energy of motion and depends on an object's mass and speed. The work-kinetic energy theorem states that the net work done on an object equals the change in its kinetic energy.
- Potential energy is the energy an object possesses due to its position or state. The work done by a constant force equals the product of force, displacement, and the cosine of the angle between them.
1. The lecture covered work, kinetic energy, and energy conservation.
2. Work is the transfer of energy via a force. It can be positive, negative, or zero depending on the angle between the force and displacement.
3. Kinetic energy is defined as 1/2 mv^2 and represents the energy of motion. The work done on an object causes a change in its kinetic energy.
This document discusses simple harmonic motion and elasticity. It covers topics like Hooke's law, springs, oscillations, energy related to springs, pendulums, damping, driven harmonic motion and resonance. Several examples are provided to illustrate concepts like calculating restoring forces on springs, determining natural frequencies, and relating stress and strain using Hooke's law.
1) The document discusses the principle of virtual work which is used to calculate deflections in statically determinate and indeterminate structures.
2) It defines virtual work and differentiates between external and internal virtual work. The principle of virtual displacement and virtual forces are also introduced.
3) The unit load method for calculating deflections is described. This involves considering a structure under a system of actual loads that produce real displacements, and a separate system of virtual loads that produce virtual displacements.
1) A student pointed out that a sailboat with a fan blowing into its sails could in fact move forward, because the air molecules bouncing off the sail would impart twice the momentum change experienced going through the fan, providing a net forward force.
2) A 2000-kg car traveling at 30 m/s collided perfectly inelastically with another 2000-kg car traveling at 10 m/s, sticking together with a final speed of 20 m/s. 20% of the initial kinetic energy was lost to heat and deformation.
3) A 16-g bullet fired into a 1.5-kg ballistic pendulum was calculated to have a speed of 45 m/s before impact by applying
1) The document discusses the work-energy principle, which states that the work done on an object equals its change in kinetic energy.
2) It provides various examples of calculating work done by different types of forces like inclined, gravitational, frictional, and spring forces.
3) Key concepts covered include the definitions of work and work done, sign conventions for different forces, and using the work-energy principle to solve problems by setting the work equal to the change in kinetic energy.
Principles of soil dynamics 3rd edition das solutions manualTuckerbly
Download at: https://goo.gl/NfK7oK
People also search:
principles of soil dynamics 3rd edition pdf
principles of soil dynamics by braja m das pdf
principles of soil dynamics das
principle of soil dynamics
soil dynamics pdf
soil dynamics books
principles of foundation engineering
The document discusses work, energy, and the conservation of mechanical energy. It defines work as the product of force and displacement, and introduces kinetic energy as the energy of motion and potential energy as stored energy due to position or force interactions. The document also explains that the total mechanical energy, which is the sum of an object's kinetic and potential energies, remains constant in an isolated system according to the law of conservation of energy.
Work is defined as a force causing displacement. James Joule discovered the relationship between work, energy, and temperature. Mechanical energy exists in kinetic and potential forms. Kinetic energy depends on an object's motion while potential energy depends on its position or elastic source. Work and energy calculations can be used to determine values like spring constant and power.
The document provides an introduction to work, energy and power in physics. It defines work as the product of the component of a force in the direction of displacement and the magnitude of displacement. Kinetic energy is defined as half the mass times the square of an object's speed. The work-energy theorem states that the change in an object's kinetic energy is equal to the work done on it by net force. Positive work is done when force and displacement are in the same direction, and negative work is done when they are in opposite directions. Power is defined as the rate of doing work or energy transfer.
This document discusses work, energy, and the conservation of mechanical energy. It defines work as the product of force and displacement, with units of joules. Kinetic energy depends on an object's mass and speed, and potential energy includes gravitational potential energy and elastic potential energy. The principle of conservation of mechanical energy states that the total mechanical energy of an isolated system remains constant over time. Examples are provided to demonstrate calculations of work, kinetic energy, potential energy, and applying the conservation of mechanical energy to solve for unknown values.
This document discusses key concepts of work, energy, and power. It defines energy as the capacity to do work and defines work in physics as the application of a force through a distance. Units of energy are joules, which equal kg-m^2/s^2. Kinetic energy is defined as 1/2mv^2 and represents the energy of motion. Gravitational potential energy is defined as mgh and represents energy stored by raising a mass against gravity. Examples are provided to illustrate kinetic energy, gravitational potential energy, and how energy is transferred and transformed between the two.
Strong Nuclear Force and Quantum Vacuum as Gravity (FUNDAMENTAL TENSOR)SergioPrezFelipe
Publication at ccsenet. Gravity explained by a new theory, ‘Superconducting String Theory (SST)’, completely opposite from current field emission based and inspired on originals string theories. Strengths are decomposed to make strings behave as one-dimensional structure with universe acting as a superconductor where resistance is near 0 and the matter moves inside. Strong nuclear force, with an attraction of 10.000 Newtons is which makes space to curve, generating acceleration, more matter more acceleration. Electromagnetic moves in 8 decimals, gravity is moved to more than 30 decimals to work as a superconductor.
- Work is done when a force causes an object to move in the direction of the force. No work is done if there is no movement.
- Work (W) is calculated as the product of the magnitude of the force (F) and the magnitude of displacement (d) in the direction of the force.
- Power is the rate at which work is done and is calculated as work (W) divided by time (t). It is measured in watts, with 1 watt equaling 1 joule per second.
The document discusses Newton's second law of motion which states that acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, and inversely proportional to the mass of the object. It provides examples of calculating the acceleration of objects given their mass and applied net force using the equation a=Fnet/m. These examples include kicking a football, lifting an elevator car with passengers, and forces acting on an object from different directions.
The document discusses impulse, momentum, and collisions. It defines impulse as the product of an average force and the time it acts, and momentum as the product of an object's mass and velocity. The impulse-momentum theorem states that impulse equals change in momentum when a net force acts. Conservation of momentum means the total momentum of an isolated system remains constant. Collisions can be elastic or inelastic, depending on whether kinetic energy is conserved.
Work, energy, and power are defined. Work is force times distance. Kinetic energy is equal to one-half mass times velocity squared. Power is the rate of doing work, defined as work per unit time. Several examples of calculating work are provided for different scenarios like lifting an object, springs, and overcoming friction. The key concepts of work, kinetic energy, and power are reviewed.
1) The document provides calculations for force (894 N), acceleration (9.93 m/s^2), kinetic energy (50J), velocity (10 m/s), momentum, work (22782.06 J), and power using various physics equations.
2) It identifies that the pairs of interacting objects in different scenarios involving a rocket are the rocket and exhaust gases, exhaust gases and air, and gravity, rocket exhaust, and friction.
3) It states that compression waves are most clearly seen in longitudinal waves like a slinky where the coils are close together.
The document discusses mechanical energy and its two forms: potential and kinetic energy. Potential energy is energy due to an object's position or state, like a raised weight or stretched spring, while kinetic energy is energy due to an object's motion. Mechanical energy, the energy due to an object's position or motion, is the sum of its potential and kinetic energies. Examples are given like a bow's potential energy being transferred to an arrow's kinetic energy on release. The relationships between work, potential and kinetic energy are also explained.
1. Work done by a constant force depends on the magnitude of the force and the displacement along the direction of the force. Work done by opposing forces is negative. Centripetal forces do no work as they are always perpendicular to motion.
2. The work-kinetic energy theorem states that the work done on an object equals its change in kinetic energy. It can be used to calculate changes in speed.
3. Gravitational potential energy is defined as mgh. The principle of conservation of mechanical energy can be used to solve problems involving changes in kinetic and potential energy in an isolated system, such as an object moving under gravity.
With this mantra success is sure to come your way. At APEX INSTITUTE we strive our best to realize the Alchemist's dream of turning 'base metal' into 'gold'.
B conservative and non conservative forcesdukies_2000
This document discusses conservative and non-conservative forces, and the principles of conservation of energy and mechanical energy. It states that for conservative forces, the total energy within a closed system remains the same, though it can transform between potential and kinetic forms. For conservative forces, the net work over a closed loop is zero, and the work is path independent. Friction is a non-conservative force where net work is done over a closed loop and more work is done over longer distances. Potential energy is the other form of energy involved in conservative systems, where the sum of potential and kinetic energy equals the total energy and changes in one form equal negative changes in the other.
Karen Adelan presented on the topic of classical mechanics and energy. Some key points:
- Energy is a conserved quantity that can change forms but is never created or destroyed. It is useful for describing motion when Newton's laws are difficult to apply.
- Kinetic energy is the energy of motion and depends on an object's mass and speed. The work-kinetic energy theorem states that the net work done on an object equals the change in its kinetic energy.
- Potential energy is the energy an object possesses due to its position or state. The work done by a constant force equals the product of force, displacement, and the cosine of the angle between them.
1. The lecture covered work, kinetic energy, and energy conservation.
2. Work is the transfer of energy via a force. It can be positive, negative, or zero depending on the angle between the force and displacement.
3. Kinetic energy is defined as 1/2 mv^2 and represents the energy of motion. The work done on an object causes a change in its kinetic energy.
1. The lecture covered work, kinetic energy, and energy conservation.
2. Work is the transfer of energy using a force. Work can be positive, negative, or zero depending on the angle between the force and displacement.
3. Kinetic energy is related to an object's motion and is calculated as K = 1/2 mv^2. Work done on an object changes its kinetic energy such that the total work is equal to the change in kinetic energy.
1) In an elastic collision between two bodies in one dimension, both linear momentum and kinetic energy are conserved.
2) By applying the laws of conservation of momentum and kinetic energy, equations relating the velocities of the bodies before and after collision can be derived.
3) These equations allow calculating the unknown velocities if the masses of the bodies and their velocities before collision are known.
This document provides an overview of key concepts related to work, energy, and power including:
- The definitions and relationships between work, kinetic energy, gravitational potential energy, and elastic potential energy.
- Conservative and non-conservative forces.
- How to calculate work done by non-conservative forces.
- The work-energy theorem and the law of conservation of energy.
- The definition of power as the rate of doing work.
Welcome to the fascinating world of Work, Energy, and Power! In the realm of physics, these concepts form the cornerstone of understanding how objects interact with each other and how energy is transformed within systems. From the motion of everyday objects to the dynamics of celestial bodies, the principles of work, energy, and power are ubiquitous, shaping the very fabric of the universe.
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The spring constant is found by k=F/x, which gives k=64,000 N/m. The work done on the spring is found using W=1/2kx^2, which is 3,200 J. The force to stretch it 1.90 m is kx or 121,600 N. The power used is the work done divided by time, which is 1,600 W.
Work, energy, and power are defined. Work is force times distance. Kinetic energy is equal to one-half mass times velocity squared. Power is the rate of doing work, defined as work per unit time. Several examples of calculating work are provided for different scenarios like lifting an object, lowering an object, and work done by springs. The concept that work is based on the force component in the direction of motion is emphasized.
2 work energy power to properties of liquidsAntony Jaison
1) Work is done when a force causes an object to be displaced. It is defined as the product of the force and displacement in the direction of the force. Work is a scalar quantity measured in joules.
2) Energy is the ability to do work and exists in kinetic and potential forms. Kinetic energy is the energy of motion and potential energy is stored energy due to an object's position or state.
3) According to the work-energy theorem, the work done on an object equals its change in kinetic energy. For a variable force, the work is calculated as the area under the force-displacement graph.
2 work energy power to properties of liquidsarunjyothi247
Work is done when a force causes an object to be displaced. Work is defined as the product of the force and displacement in the direction of the force. Kinetic energy is the energy an object possesses due to its motion. Potential energy is the energy an object possesses due to its position or state. The law of conservation of energy states that energy cannot be created or destroyed, only changed from one form to another. Elastic collisions are collisions where both momentum and kinetic energy are conserved, while inelastic collisions conserve momentum but not kinetic energy.
This document provides information about physics concepts related to work. It defines work using the formula Work = Force x Distance, and explains this definition in examples of Atlas holding up the Earth and a waiter carrying a tray. It then discusses kinetic energy and the work-energy theorem. Examples are provided for work done by gravity over a change in height, gravitational potential energy, conservation of energy, free fall, pendulums, roller coasters, springs, and multiple forces. Power is also defined and examples using kilowatt-hours are given. The document concludes with multiple choice questions testing understanding of these concepts.
This document covers concepts in one-dimensional and three-dimensional kinematics, dynamics, work, energy, momentum, rotational motion, and more. Examples are provided to demonstrate how to apply equations for instantaneous and average velocity/acceleration, projectile motion, Newton's laws, work-energy theorem, impulse-momentum, center of mass, moment of inertia, and torque. Problem-solving strategies are outlined for analyzing forces, energy, momentum, and rotational equilibrium.
The document discusses key physics concepts related to motion, forces, energy, and electricity. It defines terms like speed, velocity, acceleration, force, work, power, kinetic energy, potential energy, current, voltage, and resistance. Formulas are provided for calculating these values along with example problems and explanations of physics principles.
Explain work, energy and power. The Law of Conservation of Energy is utilized as well as conservative and non conservative systems.
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This document discusses concepts of work, energy, and power. It defines work as a force causing displacement and introduces equations to calculate work. It distinguishes between kinetic energy as the energy of motion and potential energy as stored energy due to an object's position or elastic source. Formulas are provided to calculate potential energy, kinetic energy, spring constant, and power.
The document defines work, energy, and power. It explains that work is the transfer of energy by a force acting on an object, and is calculated as work = force x distance. It also discusses different types of energy including kinetic, potential, and chemical energy. Power is defined as the rate of work done or energy transferred, and is calculated as power = work / time. Examples of calculations involving work, energy, and power are provided.
The binding of cosmological structures by massless topological defectsSérgio Sacani
Assuming spherical symmetry and weak field, it is shown that if one solves the Poisson equation or the Einstein field
equations sourced by a topological defect, i.e. a singularity of a very specific form, the result is a localized gravitational
field capable of driving flat rotation (i.e. Keplerian circular orbits at a constant speed for all radii) of test masses on a thin
spherical shell without any underlying mass. Moreover, a large-scale structure which exploits this solution by assembling
concentrically a number of such topological defects can establish a flat stellar or galactic rotation curve, and can also deflect
light in the same manner as an equipotential (isothermal) sphere. Thus, the need for dark matter or modified gravity theory is
mitigated, at least in part.
EWOCS-I: The catalog of X-ray sources in Westerlund 1 from the Extended Weste...Sérgio Sacani
Context. With a mass exceeding several 104 M⊙ and a rich and dense population of massive stars, supermassive young star clusters
represent the most massive star-forming environment that is dominated by the feedback from massive stars and gravitational interactions
among stars.
Aims. In this paper we present the Extended Westerlund 1 and 2 Open Clusters Survey (EWOCS) project, which aims to investigate
the influence of the starburst environment on the formation of stars and planets, and on the evolution of both low and high mass stars.
The primary targets of this project are Westerlund 1 and 2, the closest supermassive star clusters to the Sun.
Methods. The project is based primarily on recent observations conducted with the Chandra and JWST observatories. Specifically,
the Chandra survey of Westerlund 1 consists of 36 new ACIS-I observations, nearly co-pointed, for a total exposure time of 1 Msec.
Additionally, we included 8 archival Chandra/ACIS-S observations. This paper presents the resulting catalog of X-ray sources within
and around Westerlund 1. Sources were detected by combining various existing methods, and photon extraction and source validation
were carried out using the ACIS-Extract software.
Results. The EWOCS X-ray catalog comprises 5963 validated sources out of the 9420 initially provided to ACIS-Extract, reaching a
photon flux threshold of approximately 2 × 10−8 photons cm−2
s
−1
. The X-ray sources exhibit a highly concentrated spatial distribution,
with 1075 sources located within the central 1 arcmin. We have successfully detected X-ray emissions from 126 out of the 166 known
massive stars of the cluster, and we have collected over 71 000 photons from the magnetar CXO J164710.20-455217.
When I was asked to give a companion lecture in support of ‘The Philosophy of Science’ (https://shorturl.at/4pUXz) I decided not to walk through the detail of the many methodologies in order of use. Instead, I chose to employ a long standing, and ongoing, scientific development as an exemplar. And so, I chose the ever evolving story of Thermodynamics as a scientific investigation at its best.
Conducted over a period of >200 years, Thermodynamics R&D, and application, benefitted from the highest levels of professionalism, collaboration, and technical thoroughness. New layers of application, methodology, and practice were made possible by the progressive advance of technology. In turn, this has seen measurement and modelling accuracy continually improved at a micro and macro level.
Perhaps most importantly, Thermodynamics rapidly became a primary tool in the advance of applied science/engineering/technology, spanning micro-tech, to aerospace and cosmology. I can think of no better a story to illustrate the breadth of scientific methodologies and applications at their best.
The debris of the ‘last major merger’ is dynamically youngSérgio Sacani
The Milky Way’s (MW) inner stellar halo contains an [Fe/H]-rich component with highly eccentric orbits, often referred to as the
‘last major merger.’ Hypotheses for the origin of this component include Gaia-Sausage/Enceladus (GSE), where the progenitor
collided with the MW proto-disc 8–11 Gyr ago, and the Virgo Radial Merger (VRM), where the progenitor collided with the
MW disc within the last 3 Gyr. These two scenarios make different predictions about observable structure in local phase space,
because the morphology of debris depends on how long it has had to phase mix. The recently identified phase-space folds in Gaia
DR3 have positive caustic velocities, making them fundamentally different than the phase-mixed chevrons found in simulations
at late times. Roughly 20 per cent of the stars in the prograde local stellar halo are associated with the observed caustics. Based
on a simple phase-mixing model, the observed number of caustics are consistent with a merger that occurred 1–2 Gyr ago.
We also compare the observed phase-space distribution to FIRE-2 Latte simulations of GSE-like mergers, using a quantitative
measurement of phase mixing (2D causticality). The observed local phase-space distribution best matches the simulated data
1–2 Gyr after collision, and certainly not later than 3 Gyr. This is further evidence that the progenitor of the ‘last major merger’
did not collide with the MW proto-disc at early times, as is thought for the GSE, but instead collided with the MW disc within
the last few Gyr, consistent with the body of work surrounding the VRM.
Current Ms word generated power point presentation covers major details about the micronuclei test. It's significance and assays to conduct it. It is used to detect the micronuclei formation inside the cells of nearly every multicellular organism. It's formation takes place during chromosomal sepration at metaphase.
PPT on Direct Seeded Rice presented at the three-day 'Training and Validation Workshop on Modules of Climate Smart Agriculture (CSA) Technologies in South Asia' workshop on April 22, 2024.
ESA/ACT Science Coffee: Diego Blas - Gravitational wave detection with orbita...Advanced-Concepts-Team
Presentation in the Science Coffee of the Advanced Concepts Team of the European Space Agency on the 07.06.2024.
Speaker: Diego Blas (IFAE/ICREA)
Title: Gravitational wave detection with orbital motion of Moon and artificial
Abstract:
In this talk I will describe some recent ideas to find gravitational waves from supermassive black holes or of primordial origin by studying their secular effect on the orbital motion of the Moon or satellites that are laser ranged.
(June 12, 2024) Webinar: Development of PET theranostics targeting the molecu...Scintica Instrumentation
Targeting Hsp90 and its pathogen Orthologs with Tethered Inhibitors as a Diagnostic and Therapeutic Strategy for cancer and infectious diseases with Dr. Timothy Haystead.
The technology uses reclaimed CO₂ as the dyeing medium in a closed loop process. When pressurized, CO₂ becomes supercritical (SC-CO₂). In this state CO₂ has a very high solvent power, allowing the dye to dissolve easily.
2. Motivation to go beyond Newton’s laws
Loop the loop
R
mg
N
mg
N
mg
N
The normal force has different
direction and magnitude at every
point on the track!!
Writing and solving Newton’s laws can be a nasty experience
3. Playing around with Newton’s 2nd law.
Consider the motion of a bead of mass m on a straight wire pushed by
a constant net force Fx parallel to the wire, along a displacement Δx:
Fnet,x
m
Δx
x x
F = m aForce produces
acceleration:
2 2
x f i
2a Δx = v - v
vi vf
Acceleration produces
change in speed:
2 2x
f i
F
2 Δx = v - v
m
Change in
KINETIC ENERGY
(K) of the bead
2 2
net,x f i
1 1
F Δ x = m v - m v
2 2
WORK (W ) done
by force F over
displacement Δx
4. This is an expression of the “effectiveness” of the force.
net
W = ΔK
Of how a force applied
over a distance…
…changes something in the
system
Work
The “external agent” that
changes the amount of kinetic
energy (the state) in the system
x
W = F Δ x
Kinetic energy
The “internal” quantity (state)
of the system
21
K = mv
2
2 2
net,x f i
1 1
F Δx = mv - mv
2 2
Work/Kinetic Energy Theorem
5. m
Δx
Initial kinetic energy Ki Final kinetic energy Kf > Ki
Positive work W >
0
Energy
source
Work is energy being transferred
Fnet,x
6. Units for work and energy:
SI:
Not to be confused with:
Calorie (or food calorie) 1 Cal = 1000 cal
calorie 1 cal = 4.184 J
Kilowatt-hour KWh
Other common units:
Joule 1 J = 1 N · m
7. Energy
Many types of energy:
• kinetic energy
• electric energy
• internal –thermal- energy
• elastic energy
• chemical energy
• Etc.
Energy is transferred and
transformed from one type
to another and is never
destroyed or created.
8. What is work?
The definition of work W =F Δx corresponds to the intuitive idea of
effort:
• More massive object will require more work to get to the same
speed (from rest)
• For a given mass, getting to a higher speed (from rest) requires
more work
• If we push for a longer distance, it’s more work.
• It takes the same work to accelerate the object to the right as to
the left (both displacement and force reverse)
9. Other type of
energy
2 2
net,x f i
1 1
F Δx = mv - mv
2 2
Consider the bead again. This time, the force points in the opposite
direction:
m
Δx
vi vf
< 0
Initial kinetic
energy Ki
Final kinetic energy Kf < Ki
Negative work W <
0
Fnet,x
10. Example: Pushing a box with friction
Paul pushes a box along 10 m across the floor at a constant velocity
by exerting a force of 200N.
Work by Paul on the box: WPaul = (200 N)(10 m) = 2000 J
(energy coming from the biochemical reactions in his muscles)
Work by friction on the box: Wf = (-200 N)(10 m) = -2000 J
(released as thermal energy into the air and the floor)
FPaul
f
k
Paul k
k Paul
F - f = ma = 0 (constant speed)
f = F
WKE theorem: WPaul + Wf = 0 Kf – Ki = 0
11. Example: Free fall with WKE
2
W = ΔK
1
mgh = mv
2
A ball is dropped and hits the ground 50 m below. If the initial speed is 0 and
we ignore air resistance, what is the speed of the ball as it hits the ground?
Change in K: 2
final initial
1
ΔK = K - K = m v - 0
2
We can use kinematics or… the WKE theorem.
Work done by gravity: mgh
mg
Δr
2
v = 2gh = 2(9.8 m / s )(50 m) = 31m / s
A. 50 m/s
B. 42 m/s
C. 31 m/s
D. 23 m/s
E. 10 m/s
12. Examples of Non-Work
This person is not doing any work on the rock since the centripetal
force is perpendicular to the direction that the stone is moving.
T
13. Examples of Non-Work
FPaul
fk
mg
N
The weight of the box that Paul pushes along a horizontal surface
does no work since the weight is perpendicular to the motion
The normal by the floor does no work either.
Δx
14. Example: Systems with several objects
μk = 0.2
m2 = 5.0 kg
m1 = 3.0 kg
What is the speed of the system after box 1 has fallen for 30 cm?
Two approaches:
1. Use Newton’s laws to find acceleration; use
kinematics to find speed. (long!!)
2. Use WKE theorem (smart!)
15. μk = 0.2
m2 = 5.0 kg
m1 = 3.0 kg
What is the speed of the system after box 1 has fallen for 30 cm?
External forces doing work: m1g, fk
(The tensions are internal forces: their net work is zero)
net 1 k
W = m gx - f x
m1g
m2g
N
fk
T
T
= x
1 k 2
= m gx - μ m gx
2 2
1 2
1 1
ΔKE = m v + m v - 0
2 2
16. ( ) ( ) 2
1 k 2 1 2
1
m - μ m gx = m + m v
2
1 k 2
1 2
m - μ m
v = 2gx
m + m
m1 = 3.0 kg
m2 = 5.0 kg
μk = 0.2
x = 0.3 m
3 - 0.2 × 5
= 2(9.8)(0.3) =1.2 m / s
3 + 5
18. The story so far:
net
W = ΔK
Kinetic energy 21
K = mv
2
/ /
W = F.Δr = FΔrcosθ = F Δr
r r
Work by a constant force, along a straight path:
Work/Kinetic energy theorem
19. The journey is divided up into a series of segments over which the force is
constant.
Work by non-constant force, with straight line trajectory
An object moves along the x-axis from point x1 to x2. A non-constant force
is applied on the object. What is the work done by this force?
x1
FA
ΔxA
FB
ΔxB
FC
ΔxC
FD
ΔxD
FE
ΔxE
Ax A Bx B Cx C Dx D Ex E
W = F Δx + F Δx + F Δx + F Δx + F Δx
The total work is the sum of the works for each of the intervals:
x2
20. Fx
x
ΔxA ΔxB ΔxC ΔxD ΔxE
Work=area
FAx
FBx
FCx
FDx
FEx
This is still an approximation, because the force is not really
constant in each interval
Ax A Bx B Cx C Dx D Ex E
W = F Δx + F Δx + F Δx + F Δx + F Δx
21. F
x
Work=Area under F(x) curve
x1 x2
The calculation is really good in the limit where the intervals are very small,
the sum becomes an integral:
ò
2
1
x
xx
W = F (x)dx
Work by non-constant force, with
straight line trajectory
22. Springs
Hooke’s law: The force exerted by a spring is proportional to the distance the
spring is compressed/stretched from the relaxed position.
Δx = 0 Δx = 0 F = 0
Fx = −k Δx k = spring constant
Δx
Δx > 0 F < 0
F
Δx
Δx < 0 F > 0
F
23. Stretching a Spring
What is the work done by the spring on the block as the tip is pulled
from x1 to x2?
Relaxed position (Define x = 0 as the
position of the end of the relaxed
spring. Then F = -kx)
F1 = -kx1
x1
x2
F2 = -kx2
Varying force: an
integral is required.
24. 22 2
1 1 1
2
by spring
1
( )
2
xx x
x x x
W F x dr kxdx kx
ù æ ö
= × = - = - ç ÷ú
û è ø
ò ò
2 2
2 1
1 1
= - kx - kx
2 2
r r
What is the work done by a spring as the tip is pulled from x1 to x2?
Fx
x
x1 x2
work
Fx = kx
If x2 > x1 (stretch),
Wexternal > 0
We are adding energy
to the spring
A stretched or
compressed spring
stores energy (elastic
energy).
2 2
external by spring 2 1
1 1
W = -W = kx - kx
2 2
What is the work done on the spring as the tip is pulled from x1 to x2?
25. Example: Box and spring
A box of mass m = 25 kg slides on a horizontal frictionless surface with an
initial speed v0 = 10 m/s. How far will it compress the spring before coming to
rest if k = 3000 N/m?
A. 0.50 m B. 0.63 m C. 0.75 m
D. 0.82 m E. 0.91 m
x
v = 10 m/s
m = 25 kg k = 3000 N/m
26. W = ΔKE
2 2
2 2
1 1
kx = mv
2 2
mv (25 kg)(10 m / s)
x = = = 0.91 m
k 3000 N / m
F
x
ò
x
2
0
1
W = - kxdx = - kx
2
Use the work-kinetic energy theorem:
21
ΔKE = 0 - mv
2
x
v = 10 m/s
m = 25 kg k = 3000 N/m
DEMO:
Horizontal
spring
Answer E
27. Work on curved trajectories
A particle moves from A to B along this
path while a varying force acts upon it.
l
rr
dFdW ×=
If the intervals are very small, the
trajectory is straight and the force
is constant, so the work done by
a force is:
Again, we can approximate the work
by considering breaking it into small
displacements .l
r
d
We need to add –integrate-
all the contributions: ® ò
B
A B A
W = F.dl
rr
x A
B x
F dl
28. The line integral
A (initial point)
B (final point)
For three different paths to
move from A to B, a force
will in general do different
works!
path
® ò
B
A B A
W = F.dl
rr
29. rim
bottom
mg
θ
θ
Example: Spherical bowl
Find the work done by gravity on a pebble of mass m as it rolls from the
rim to the bottom of a spherical bowl of radius R.
dl
ò
ò
ò
bottom
top
θ=π / 2
θ=0
π / 2
0
π / 2
0
W = mg × dl
= mg Rdθ cosθ
= mgR cosθ dθ
= mgR sinθ
= mgR
rr
= R dθ
30. Power
Instantaneous power:
dW F × dr
P = P = = F × v
dt dt
r r r r
average average average
W F × Δr
P = P = = F × v
Δt Δt
r r r r
Average power:
31. Units of power
SI: Watt 1 W = 1 J/s
Other: Horsepower 1 hp = 746 W
Kilowatt-hour (kWh) is a unit of energy or work:
61000 W 3600 s
1 kWh =1 kW × h = 3.6 ×10 Ws
1 kW 1 h
J
32. Example: Bullet and wood block
A bullet of mass 10–3 kg traveling at a speed of 300 m/s strikes a
block of wood. It embeds a distance of 1 cm. How much force does
the wood exert on the bullet as it slows it down?
Identify
Let us assume that the force is constant (or that we will find the
average force)
The force must do enough negative work on the bullet to render
its kinetic energy 0.
33. f i
2
2
i
f i i
W = ΔK
FΔx = K - K
1
0 - mvK - K mv2F = = = -
Δx Δx 2Δx
WKE theorem:Setup
Execute
2 -3 2
i
-2
mv (10 kg)(300 m / s)
F = - = - = -4500 N
2Δx 2(10 m)
The wood exerts a force of –4500N on the bullet. It should be
negative (opposite to Δx) because it must decelerate the bullet.
Evaluate
34. EXAMPLE: Hammer
A hammer slides along 10 m down a 30° inclined roof and off into the
yard, which is 7 m below the roof edge. Right before it hits the ground,
its speed is 14.5 m/s. What is the coefficient of kinetic friction between
the hammer and the roof?
Δx = 10 m
h = 7 m
v = 14.5 m/s
30
This can be solved using Newton’s
laws and kinematics, but it’s
looooooooooooooooooooooooong.
35. ( ) ( )
( ) ( )
2
net on the roof projectile
k
k
1
W = W + W ΔKE = mv - 0
2
= mgh' - f Δx + mgh
= mg Δxsinθ - μ mgcosθ Δx + mgh
Δx = 10 m
h = 7 m
v = 14.5 m/s
h’ θ
Normal by an inclineh’ = Δx sinθ
( )2
k
1
mv = mgΔxsinθ - μ mgcosθ Δx + mgh
2
( ) ( )2 2
k
2g h + Δxsinθ - v 2(9.8) 7 +10sin30° -14.5
μ = = =1.5
2gΔxcosθ 2(9.8)10cos30°
37. Work done by gravityWork done by gravity
A block of mass m is lifted from the floor (A) to a table (B) through two
different trajectories. Find the work done by gravity.
Δy
mg
A
B
Δr3
Δr2
Δr1
Δr
( )
1 2 3
1 2 3
W = mg × Δr = -mgΔy
W = mg × Δr + mg × Δr + mg × Δr
= mg × Δr + Δr + Δr
= mg × Δr
= -mgΔy
r r
r r r r r r
r r r r
r r
Work by gravity does not
depend on the path
38. Gravitational potential energy
The work done by gravity does not depend on the path, it only depends on
the vertical displacement Δy, or on the initial and final y:
W = -mgΔy
U = potential energy
We can ALWAYS write this work as (minus) the change in some function
U(r) that depends on position (not on path):
( )f i
W = - U - U = -ΔU
U = mgy + constantGravitational potential energy:
There is always room for an arbitrary constant,
because what matters is ΔU
39. Can work always be written in terms of a potential
energy change?
NO!
Example: A box is dragged along a rough horizontal surface
through two paths AD and ABCD:
A
B C
D
friction,AD k
W = -df
The work done by friction CANNOT
be written as a potential difference.
Does not depend
on initial and final
points only.
friction,ABCD k
W = -3df
40. Conservative and non-conservative forces
The work done by a conservative force does not depend on
the trajectory.
• A potential energy function can be defined.
• The work done by a non-conservative force depends on the
trajectory.
• A potential energy function cannot be defined.
Non-conservative force = force that is not conservative.
Examples: Kinetic friction
Examples: Gravity, spring
41. Conservation of Mechanical Energy
In a system where only conservative forces are doing work, we can
rewrite the WKE theorem:
KW D=net
UW D-=net
KU D=D- 0)( =+D UK
Definition of Mechanical Energy: UKE +=
Under the conditions above,
mechanical energy is conserved:
finalinitial
or0 EEE ==D
42. 2
W = ΔK
1
mgh = mv
2
v = 2gh
A ball is dropped from a height h. If the initial speed is 0 and we ignore air
resistance, what is the speed of the ball as it hits the ground?
We can use kinematics or… the WKE theorem… or conservation of energy.
Work done by
gravity: mgh
mg
Δr
initial final
initial initial final final
2
2
E = E
K +U = K + U
1
0 + mgh = mv + 0
2
1
mgh = mv
2
v = 2gh
The only force doing work is gravity, so
mechanical energy is conserved.
Choice: U = 0
at ground level
WKE Conservation of energy
Example: Free fall
43. Two ways of looking at the same problem.
1. Lifting: Wnet = Wgravity + Wperson = 0 (vinitial = vfinal = 0)
Drop: Wnet =Wgravity = mgh > 0 (thus K increases)
An object of mass m is lifted by a person from the floor to
height h and dropped.
1. Lifting: Wperson > 0. Person is adding energy which is
stored as gravitational potential energy mgh > 0 .
Drop: The potential energy is converted into kinetic
energy (thus K increases)
44. The really nice thing is, we can apply the same
thing to any “incline”:
h
Turn-around
point: where
K = 0
E K U
DEMO:
Wavy trackE K U
E K U
v = 0
45. Example: Loop-the-loop
A. 1.5R
B. 2.0R
C. 2.5R
D. 3.0R
E. 4.0R
A cart is released from height h in a roller coaster with a
loop of radius R. What is the minimum h to keep the cart
on the track?
h
R
Impossible, h must be
at least 2R
Cool
…
46. h
R
Aaaah…!!!!
A
B
Point B is the toughest point. What is the speed there?
(Eqn. 1)
A B
2
B
B
E = E
1
mgh+ 0 = mg2R + mv
2
v = 2g(h - 2R)
47. h
R
Aaaah…!!!!
A
B
In order not to fall (ie, to keep the circular trajectory),
the forces at B must provide the appropriate radial
acceleration: 2
B
v
mg + N = m
R
mg
Nby track
The minimum velocity is fixed by N = 0:
Þ
2
B,min
B,min
v
mg = m v = gR
R
(Eqn. 2)
48. Let us put equations 1 and 2 together:
B B,min
v = 2g(h - 2R) v = gR
The minimum height is given by:
min
gR = 2g(h - 2R)
min
min
R = 2h - 4R
5
h = R
2
Answer C
49. Appendix: Loop-the-loop with
Newton’s laws
A cart is released from height h in a roller
coaster with a loop of radius R. What is the
minimum h to keep the cart on the track?
h
R
Cool
…
50. h
mg
N
Part 1: The straight section
mgsinθ = ma
a = gsinθ
θ
x
Speed at the bottom:
2
bottom
bottom
v = 2ax
h
= 2a
sinθ
h
v = 2 gsinθ
sinθ
= 2gh
51. R
Part 2: The circular section
mg
N
θ
θ
In the radial direction, at
any given θ:
2
v
N - mgcosθ = m
R
In the tangential direction,
at any given θ: t
mgsinθ = ma
This is circular
motion with a non-
uniform angular
acceleration!
52. 2
N - mgcosθ = mRω
gsinθ = Rα
In terms of angular quantities:
æ ö
ç ÷
è ø
2
dθ
N - mgcosθ = mR
dt2
2
d θ
gsinθ = R
dt
Differential equations for θ (at
least not coupled)
(1)
(2)
53. dω g
ω = sinθ
dθ R
2
2
d θ dω dω dθ dω
On equation (2),use this trick : = = = ω
dt dθ dt dθdt
g
ω dω = sinθdθ
R
( ) ( )2 2
0
1 g
ω - ω = cosθ - cos0°
2 R
where ω0 is the angular speed
at the bottom, see part 1:
bottom
0
v 2gh
ω = =
R R
( )
æ ö
ç ÷
è ø
2
2
2gh 2g 2g h
ω = + cosθ -1 = + cosθ -1
R R RR
integrate
(now it’s a first order differential
equation)
54. At the top, θ = 180º:
æ ö
ç ÷
è ø
2
top
2g h
ω = - 2
R R
From equation (1), and for θ = 180º:
2
top
N + mg = mRω
Minimum speed ⇒ N = 0
2
top,min
mg = mRω
(3)
(4)
(3) and (4):
æ ö
ç ÷
è ø
min
h2g g
- 2 =
R R R m in
5
h = R
2
55. EXAMPLE: Pendulum
Consider a pendulum of length l and mass m, that is released
from rest at an angle θ0.
a. What is the maximum angle that the pendulum will reach
on the other side?
b. What is the maximum speed of the pendulum?
θ0
m l
56. Only weight is doing work, so it’s a situation where
mechanical energy E = KE +U is conserved.
L
m
θ0
U µ y
y
KE=0
UMAX
UMIN
KEMAX
θ0 KE=0
UMAX
So the angle
on the other
side is also θ0.
57. θ0 θ0
E K U E K U
E K U
E K U
E K U
Potential energy U is transformed into kinetic energy K. And
viceversa.
58. θ0
m L
Let’s take U(θ = 0) = 0
⇒ C = 0
U = mgL(1-cosθ)
U = mgy +
C
y
Lcosθ
= mgL(1-cosθ) + C
θ
L
L
L-Lcosθ
59. θ0
m
L
E = mgL(1-cosθ) + mv2/2 = constant
At the bottom: E = 0 + mv2/2
Initially: E = mgL(1-cosθ0) + 0
0 + mv2/2 = mgL(1-cosθ0) + 0
0
v = 2gL(1 - cosθ )