1) Angular kinematics describes motion that involves rotation, such as the movement of body segments. It includes concepts like angular displacement, velocity, and speed.
2) Key concepts in angular kinematics include computing angular quantities from changes in angular position over time, using degrees and radians as units of angle, and determining average versus instantaneous angular velocity.
3) Joint angles are relative angles between adjacent body segments and are important for analyzing human movement.
This document discusses linear and angular motion concepts including:
1) The relationship between linear and angular velocity for rotating bodies
2) Computing tangental and radial acceleration of rotating bodies
3) Analyzing general motion involving combinations of linear and angular movement
4) Methods for measuring kinematic quantities such as velocity and acceleration.
This document covers topics in circular motion, gravitation, and rotational dynamics including:
- Definitions of radian, angular displacement, average angular speed, and average angular acceleration.
- Centripetal acceleration and the forces that provide the centripetal force for circular motion.
- Newton's law of universal gravitation and applications including weighing Earth and escape speeds.
- Motion of satellites in orbit and the relationship between orbital radius, speed, and period as described by Kepler's laws of planetary motion.
- Torque as the tendency of a force to cause rotation, defined as the product of the force and the lever arm distance.
This chapter discusses rotational kinematics and the relationships between linear and rotational motion. Key concepts covered include angular displacement, velocity, and acceleration and how to define and calculate them. Equations are provided relating rotational parameters like displacement, velocity, and acceleration to their linear motion counterparts using variables like radius and arc length. Examples are given calculating values for various rotational motion situations. The chapter aims to help students understand and apply concepts of rotational kinematics.
(a) If speed doubles, centripetal force must quadruple. Radius must halve to maintain same centripetal force, so smallest radius would be r/2.
(b) If mass doubles, centripetal force must double to provide the same centripetal acceleration. Radius must halve again to maintain the doubled force, so smallest radius would be r/4.
The document discusses internal and external torques, the three laws of angular motion, and key concepts related to rotational dynamics including:
1) Internal torque is applied within a system while external torque is applied across the system boundary.
2) The three laws of angular motion describe how torques cause changes in angular velocity and acceleration according to an object's moment of inertia.
3) Key concepts like angular impulse, work, power, and kinetic energy can be analyzed similarly to linear motion but involve torque, angular velocity/acceleration, and moment of inertia rather than force and linear velocity/acceleration.
Here are the key points about momentum and impulse:
- Momentum is the product of an object's mass and velocity. It represents the amount of motion an object has.
- Impulse is the product of force and the time over which it acts. It represents the change in an object's momentum due to a force.
- Impulse and change in momentum are directly related - a large impulse (large force or long duration) results in a large change in momentum.
- Both momentum and impulse are vector quantities, having both magnitude and direction associated with the motion or force.
So in summary, momentum describes the amount of motion, while impulse describes the force applied to change an object's motion and momentum.
Okay, let's break this down step-by-step:
1) Resolve the initial velocity vector into its horizontal (Vx) and vertical (Vy) components using trigonometry:
Vx = Vo cosθ
Vy = Vo sinθ
2) The horizontal component Vx remains constant.
3) The vertical component Vy is accelerated by gravity. We can use the kinematic equations:
y = Yo + Vyot + 1/2at2
Vyo = initial vertical velocity
a = acceleration due to gravity (g)
4) To get the total displacement, we use Pythagorean theorem:
x2 + y2 = r2
Where r
1) Linear and angular kinetics relate external forces/torques to inertia, displacement/angular displacement, velocity/angular velocity, and acceleration/angular acceleration respectively.
2) Moment of inertia represents an object's resistance to changes in angular motion and depends on the object's mass distribution and the axis of rotation.
3) Angular momentum is the product of an object's moment of inertia and angular velocity, and is conserved if no external torque is applied to the system.
This document discusses linear and angular motion concepts including:
1) The relationship between linear and angular velocity for rotating bodies
2) Computing tangental and radial acceleration of rotating bodies
3) Analyzing general motion involving combinations of linear and angular movement
4) Methods for measuring kinematic quantities such as velocity and acceleration.
This document covers topics in circular motion, gravitation, and rotational dynamics including:
- Definitions of radian, angular displacement, average angular speed, and average angular acceleration.
- Centripetal acceleration and the forces that provide the centripetal force for circular motion.
- Newton's law of universal gravitation and applications including weighing Earth and escape speeds.
- Motion of satellites in orbit and the relationship between orbital radius, speed, and period as described by Kepler's laws of planetary motion.
- Torque as the tendency of a force to cause rotation, defined as the product of the force and the lever arm distance.
This chapter discusses rotational kinematics and the relationships between linear and rotational motion. Key concepts covered include angular displacement, velocity, and acceleration and how to define and calculate them. Equations are provided relating rotational parameters like displacement, velocity, and acceleration to their linear motion counterparts using variables like radius and arc length. Examples are given calculating values for various rotational motion situations. The chapter aims to help students understand and apply concepts of rotational kinematics.
(a) If speed doubles, centripetal force must quadruple. Radius must halve to maintain same centripetal force, so smallest radius would be r/2.
(b) If mass doubles, centripetal force must double to provide the same centripetal acceleration. Radius must halve again to maintain the doubled force, so smallest radius would be r/4.
The document discusses internal and external torques, the three laws of angular motion, and key concepts related to rotational dynamics including:
1) Internal torque is applied within a system while external torque is applied across the system boundary.
2) The three laws of angular motion describe how torques cause changes in angular velocity and acceleration according to an object's moment of inertia.
3) Key concepts like angular impulse, work, power, and kinetic energy can be analyzed similarly to linear motion but involve torque, angular velocity/acceleration, and moment of inertia rather than force and linear velocity/acceleration.
Here are the key points about momentum and impulse:
- Momentum is the product of an object's mass and velocity. It represents the amount of motion an object has.
- Impulse is the product of force and the time over which it acts. It represents the change in an object's momentum due to a force.
- Impulse and change in momentum are directly related - a large impulse (large force or long duration) results in a large change in momentum.
- Both momentum and impulse are vector quantities, having both magnitude and direction associated with the motion or force.
So in summary, momentum describes the amount of motion, while impulse describes the force applied to change an object's motion and momentum.
Okay, let's break this down step-by-step:
1) Resolve the initial velocity vector into its horizontal (Vx) and vertical (Vy) components using trigonometry:
Vx = Vo cosθ
Vy = Vo sinθ
2) The horizontal component Vx remains constant.
3) The vertical component Vy is accelerated by gravity. We can use the kinematic equations:
y = Yo + Vyot + 1/2at2
Vyo = initial vertical velocity
a = acceleration due to gravity (g)
4) To get the total displacement, we use Pythagorean theorem:
x2 + y2 = r2
Where r
1) Linear and angular kinetics relate external forces/torques to inertia, displacement/angular displacement, velocity/angular velocity, and acceleration/angular acceleration respectively.
2) Moment of inertia represents an object's resistance to changes in angular motion and depends on the object's mass distribution and the axis of rotation.
3) Angular momentum is the product of an object's moment of inertia and angular velocity, and is conserved if no external torque is applied to the system.
This chapter discusses dynamic engineering systems including uniform acceleration, energy transfer through various forms like potential and kinetic energy, and oscillating mechanical systems. It covers concepts like Newton's laws of motion, conservation of energy, and how energy is transferred and stored in linear and rotating systems, as well as damped oscillatory motion. Simple harmonic motion of linear and transverse systems is also qualitatively examined.
Here are the solutions to the simple harmonic motion problems:
1. Amplitude = 20 cm
Frequency = 31.4 rad/s
Period = 2π/31.4 = 0.2 s
2. Maximum displacement = 50 cm
Maximum velocity = 1000 cm/s (20π rad/s)
Maximum acceleration = 40000 cm/s^2 (400π^2 rad^2/s^2)
Number of oscillations in 5 s = 5 * 20π = 100
3. Displacement x(t) = 20 cos(2πt/0.5) cm
Velocity v(t) = -40π sin(2πt/0.5) cm
1. The document defines key physics concepts related to forces and motion such as distance, displacement, speed, velocity, acceleration, deceleration, and their relationships.
2. Examples are provided to illustrate the concepts and distinguish between distance and displacement, as well as speed and velocity.
3. Key points are summarized such as constant acceleration resulting in increasing velocity over time, while deceleration decreases velocity over time. Zero acceleration corresponds to constant velocity.
This document defines radians as a unit of measuring angles, where the radian measure of an angle is defined as the arc length of a unit circle subtended by the angle divided by the radius. It discusses converting between degree and radian measures, defines quadrantal angles in radians, and introduces the concept of coterminal angles which have the same terminal side.
1. The document discusses converting between degrees and radians using proportional relationships involving the circumference of a circle.
2. It also explains how to find the arc length and sector area of a circle using relationships involving the central angle, radius, and total circumference/area.
3. Examples are provided for finding the arc length of a 45° central angle and the area of a sector with a central angle of 60°.
This document discusses uniform circular motion and centripetal force. It defines key terms like centripetal acceleration, frequency, period, and centripetal force. It also discusses examples of calculating centripetal acceleration for objects moving in circular motion, like a revolving ball or the moon orbiting Earth. Additionally, it covers banked tracks and how banking an angle of a curved road can help cars move around the curve without skidding.
1. The document introduces gyroscopic and precessional motion, explaining that when a spinning body moves along a curved path, gyroscopic forces act on it.
2. It defines key terms like gyroscopic couple, precessional motion, and axis of precession. The gyroscopic couple arises due to a change in angular momentum when the axis of spin rotates about the axis of precession.
3. It gives the example of an airplane in a left turn. The reactive gyroscopic couple acts in the anticlockwise direction on the airplane, causing its nose to rise and tail to dip during the left turn.
This document contains a lecture on circular motion presented by Prof. Mukesh N. Tekwani. It discusses key concepts related to circular motion including:
- The relationship between linear velocity and angular velocity
- Centripetal force and that it is required for circular motion
- Examples of centripetal force in different circular motion situations
- Radial and tangential acceleration
- Banking of roads and how banking provides the necessary centripetal force for vehicles to travel in a circle.
1) A marine gyrocompass uses a freely-spinning gyroscope to determine direction based on the principles of angular momentum and the earth's constant rotation.
2) A gyroscope has three degrees of freedom - it can spin about its axis and tilt or turn in horizontal and vertical planes. The earth acts like a giant free gyroscope due to its mass, high-speed rotation, and lack of friction in space.
3) The gyroscope's angular momentum and inertia cause it to resist changes to its axis of spin, allowing it to maintain a fixed direction in space independent of the ship's movements. This gyroscopic property is used to determine true north.
gyroscope is a chapter of theory of machine. You can easily understand concepts of gyroscope in my ppt. All concepts are with suitable examples and graphics.
saurabh.rana2829@gmail.com
Principle of Circular Motion - Physics - An Introduction by Arun Umraossuserd6b1fd
The document discusses circular motion, including angular velocity, centripetal force, components of circular motion, and motion in both horizontal and vertical planes. It defines key terms like angular displacement, angular velocity, tangential velocity, centripetal acceleration, and centrifugal force. Equations are provided for these quantities. Circular motion concepts are applied to examples like stability of vehicles on banked roads and vertical circular motion with a string.
This document provides an overview of simple harmonic motion and waves. It begins by defining simple harmonic motion and providing examples of objects exhibiting SHM, such as a mass attached to a spring, a ball in a bowl, and a pendulum. It then discusses damped oscillations and how friction reduces amplitude over time. Next, it introduces the topic of wave motion, distinguishing between mechanical and electromagnetic waves. It defines key wave properties and concepts. The document concludes by describing experiments that can be performed to demonstrate water and rope waves.
The document discusses periodic motion and simple harmonic motion (SHM). It provides examples of objects that exhibit periodic motion which may or may not be SHM. SHM occurs when the net force on an object is directly proportional to the object's displacement from equilibrium and acts to restore the object to equilibrium. Examples of SHM include a pendulum with small angular displacement and a loaded spring oscillating about its equilibrium position. The document defines terms related to SHM like period, frequency, amplitude, displacement, angular frequency, phase and phase difference. It also provides examples of free oscillations that are SHM.
The document lists 24 outline maps covering various regions of the world including continents, hemispheres, countries and borders. The regions included are:
- Asia
- Africa
- Antarctica
- Australia
- Eastern Hemisphere
- Europe
- Latin America
- Mediterranean
- Middle America
- North America
- South America
- Southeast Asia
- U.S.-Mexican Border
- Western Hemisphere
- Former Soviet Union
- Middle East
- Pacific Rim
- South Central Asia
- United States
- Mexico
- World
This document contains 6 problems about circular motion. Problem 1 discusses a car rounding a bend and losing centripetal force when hitting an oil slick, causing it to travel in a straight line. Problem 2 calculates frequency and period from revolutions and time. Problem 3 calculates speed and centripetal acceleration for a fairground ride. Problem 4 does similar calculations for a mass whirled in a circle. Problems 5 and 6 calculate forces, speeds, and accelerations for carts on a circular track and an ice skater spun by another skater.
Vectors have both magnitude and direction, while scalars have only magnitude. Vectors are represented by arrows and their magnitude and direction can be shown using coordinate axes. Vectors can be added or subtracted by using the parallelogram method or by calculating the resultant vector's magnitude and direction. Components break vectors down into orthogonal x and y components to allow vectors to be added that are not at right angles.
This Unit is rely on introduction to Simple Harmonic Motion. the contents was prepared using the Curriculum of NTA level 4 at Mineral Resources Institute- Dodoma.
This document summarizes key concepts about uniform circular motion including:
- Radians are the SI unit for measuring angles where 1 radian is the central angle that spans an arc equal to the circle's radius.
- Formulas relate angular quantities like speed (ω) and displacement (θ) to linear quantities like speed (v) and arc length (s) using the radius (R).
- Centripetal force (Fc) is required to cause circular motion and is given by Fc = Mv2/R, where M is the object's mass and v is its speed.
- Banked roads allow vehicles to safely take curved portions faster by providing tilt that replaces needed friction with
1. The document discusses various types of errors that can occur in marine gyrocompasses, including latitude error, course and speed error, and ballistic deflection.
2. Latitude error, also called damping or settling error, causes the gyro spin axis to settle slightly off true north due to eccentricities in the damping mechanism. This introduces a small error that can be calculated based on latitude.
3. Course and speed error, also called steaming error, occurs because the gyro senses the combined rotation of the Earth and ship's movement, not just Earth's rotation. This introduces an error that depends on latitude, course, and speed.
4. Ballistic deflection is an error caused by accelerations from changes
Torque is the rotational counterpart of force that causes an object to rotate about its axis. Torque is calculated as the product of the lever arm, or distance from the axis of rotation, and the applied force. A seesaw provides an example where balancing requires the right combination of mass and distance from the center to equalize the torque. Torque is calculated in the example of a 500 Newton force applied to a 25 cm wrench at a right angle, yielding a torque of 125 Newton-meters.
This document introduces rotational motion and defines key terms like angular displacement, angular velocity, and angular acceleration. It discusses how to describe circular motion using angles in radians and convert between linear and angular quantities using relationships like angular velocity equals linear velocity divided by radius. Examples are provided to demonstrate calculating angular displacement from degrees or revolutions traveled, converting between angular and linear speed, and solving kinematic equations for rotational systems.
Angular acceleration is the rate of change of angular velocity over time. It is calculated as the change in angular velocity divided by the change in time. Angular acceleration, like linear acceleration, can be constant or varying. For constant angular acceleration, the angular kinematic equations relating angular displacement, velocity, acceleration, and time can be used. Understanding the relationships between linear and angular quantities like distance, speed, and acceleration is important when analyzing rotational or spinning motion.
This chapter discusses dynamic engineering systems including uniform acceleration, energy transfer through various forms like potential and kinetic energy, and oscillating mechanical systems. It covers concepts like Newton's laws of motion, conservation of energy, and how energy is transferred and stored in linear and rotating systems, as well as damped oscillatory motion. Simple harmonic motion of linear and transverse systems is also qualitatively examined.
Here are the solutions to the simple harmonic motion problems:
1. Amplitude = 20 cm
Frequency = 31.4 rad/s
Period = 2π/31.4 = 0.2 s
2. Maximum displacement = 50 cm
Maximum velocity = 1000 cm/s (20π rad/s)
Maximum acceleration = 40000 cm/s^2 (400π^2 rad^2/s^2)
Number of oscillations in 5 s = 5 * 20π = 100
3. Displacement x(t) = 20 cos(2πt/0.5) cm
Velocity v(t) = -40π sin(2πt/0.5) cm
1. The document defines key physics concepts related to forces and motion such as distance, displacement, speed, velocity, acceleration, deceleration, and their relationships.
2. Examples are provided to illustrate the concepts and distinguish between distance and displacement, as well as speed and velocity.
3. Key points are summarized such as constant acceleration resulting in increasing velocity over time, while deceleration decreases velocity over time. Zero acceleration corresponds to constant velocity.
This document defines radians as a unit of measuring angles, where the radian measure of an angle is defined as the arc length of a unit circle subtended by the angle divided by the radius. It discusses converting between degree and radian measures, defines quadrantal angles in radians, and introduces the concept of coterminal angles which have the same terminal side.
1. The document discusses converting between degrees and radians using proportional relationships involving the circumference of a circle.
2. It also explains how to find the arc length and sector area of a circle using relationships involving the central angle, radius, and total circumference/area.
3. Examples are provided for finding the arc length of a 45° central angle and the area of a sector with a central angle of 60°.
This document discusses uniform circular motion and centripetal force. It defines key terms like centripetal acceleration, frequency, period, and centripetal force. It also discusses examples of calculating centripetal acceleration for objects moving in circular motion, like a revolving ball or the moon orbiting Earth. Additionally, it covers banked tracks and how banking an angle of a curved road can help cars move around the curve without skidding.
1. The document introduces gyroscopic and precessional motion, explaining that when a spinning body moves along a curved path, gyroscopic forces act on it.
2. It defines key terms like gyroscopic couple, precessional motion, and axis of precession. The gyroscopic couple arises due to a change in angular momentum when the axis of spin rotates about the axis of precession.
3. It gives the example of an airplane in a left turn. The reactive gyroscopic couple acts in the anticlockwise direction on the airplane, causing its nose to rise and tail to dip during the left turn.
This document contains a lecture on circular motion presented by Prof. Mukesh N. Tekwani. It discusses key concepts related to circular motion including:
- The relationship between linear velocity and angular velocity
- Centripetal force and that it is required for circular motion
- Examples of centripetal force in different circular motion situations
- Radial and tangential acceleration
- Banking of roads and how banking provides the necessary centripetal force for vehicles to travel in a circle.
1) A marine gyrocompass uses a freely-spinning gyroscope to determine direction based on the principles of angular momentum and the earth's constant rotation.
2) A gyroscope has three degrees of freedom - it can spin about its axis and tilt or turn in horizontal and vertical planes. The earth acts like a giant free gyroscope due to its mass, high-speed rotation, and lack of friction in space.
3) The gyroscope's angular momentum and inertia cause it to resist changes to its axis of spin, allowing it to maintain a fixed direction in space independent of the ship's movements. This gyroscopic property is used to determine true north.
gyroscope is a chapter of theory of machine. You can easily understand concepts of gyroscope in my ppt. All concepts are with suitable examples and graphics.
saurabh.rana2829@gmail.com
Principle of Circular Motion - Physics - An Introduction by Arun Umraossuserd6b1fd
The document discusses circular motion, including angular velocity, centripetal force, components of circular motion, and motion in both horizontal and vertical planes. It defines key terms like angular displacement, angular velocity, tangential velocity, centripetal acceleration, and centrifugal force. Equations are provided for these quantities. Circular motion concepts are applied to examples like stability of vehicles on banked roads and vertical circular motion with a string.
This document provides an overview of simple harmonic motion and waves. It begins by defining simple harmonic motion and providing examples of objects exhibiting SHM, such as a mass attached to a spring, a ball in a bowl, and a pendulum. It then discusses damped oscillations and how friction reduces amplitude over time. Next, it introduces the topic of wave motion, distinguishing between mechanical and electromagnetic waves. It defines key wave properties and concepts. The document concludes by describing experiments that can be performed to demonstrate water and rope waves.
The document discusses periodic motion and simple harmonic motion (SHM). It provides examples of objects that exhibit periodic motion which may or may not be SHM. SHM occurs when the net force on an object is directly proportional to the object's displacement from equilibrium and acts to restore the object to equilibrium. Examples of SHM include a pendulum with small angular displacement and a loaded spring oscillating about its equilibrium position. The document defines terms related to SHM like period, frequency, amplitude, displacement, angular frequency, phase and phase difference. It also provides examples of free oscillations that are SHM.
The document lists 24 outline maps covering various regions of the world including continents, hemispheres, countries and borders. The regions included are:
- Asia
- Africa
- Antarctica
- Australia
- Eastern Hemisphere
- Europe
- Latin America
- Mediterranean
- Middle America
- North America
- South America
- Southeast Asia
- U.S.-Mexican Border
- Western Hemisphere
- Former Soviet Union
- Middle East
- Pacific Rim
- South Central Asia
- United States
- Mexico
- World
This document contains 6 problems about circular motion. Problem 1 discusses a car rounding a bend and losing centripetal force when hitting an oil slick, causing it to travel in a straight line. Problem 2 calculates frequency and period from revolutions and time. Problem 3 calculates speed and centripetal acceleration for a fairground ride. Problem 4 does similar calculations for a mass whirled in a circle. Problems 5 and 6 calculate forces, speeds, and accelerations for carts on a circular track and an ice skater spun by another skater.
Vectors have both magnitude and direction, while scalars have only magnitude. Vectors are represented by arrows and their magnitude and direction can be shown using coordinate axes. Vectors can be added or subtracted by using the parallelogram method or by calculating the resultant vector's magnitude and direction. Components break vectors down into orthogonal x and y components to allow vectors to be added that are not at right angles.
This Unit is rely on introduction to Simple Harmonic Motion. the contents was prepared using the Curriculum of NTA level 4 at Mineral Resources Institute- Dodoma.
This document summarizes key concepts about uniform circular motion including:
- Radians are the SI unit for measuring angles where 1 radian is the central angle that spans an arc equal to the circle's radius.
- Formulas relate angular quantities like speed (ω) and displacement (θ) to linear quantities like speed (v) and arc length (s) using the radius (R).
- Centripetal force (Fc) is required to cause circular motion and is given by Fc = Mv2/R, where M is the object's mass and v is its speed.
- Banked roads allow vehicles to safely take curved portions faster by providing tilt that replaces needed friction with
1. The document discusses various types of errors that can occur in marine gyrocompasses, including latitude error, course and speed error, and ballistic deflection.
2. Latitude error, also called damping or settling error, causes the gyro spin axis to settle slightly off true north due to eccentricities in the damping mechanism. This introduces a small error that can be calculated based on latitude.
3. Course and speed error, also called steaming error, occurs because the gyro senses the combined rotation of the Earth and ship's movement, not just Earth's rotation. This introduces an error that depends on latitude, course, and speed.
4. Ballistic deflection is an error caused by accelerations from changes
Torque is the rotational counterpart of force that causes an object to rotate about its axis. Torque is calculated as the product of the lever arm, or distance from the axis of rotation, and the applied force. A seesaw provides an example where balancing requires the right combination of mass and distance from the center to equalize the torque. Torque is calculated in the example of a 500 Newton force applied to a 25 cm wrench at a right angle, yielding a torque of 125 Newton-meters.
This document introduces rotational motion and defines key terms like angular displacement, angular velocity, and angular acceleration. It discusses how to describe circular motion using angles in radians and convert between linear and angular quantities using relationships like angular velocity equals linear velocity divided by radius. Examples are provided to demonstrate calculating angular displacement from degrees or revolutions traveled, converting between angular and linear speed, and solving kinematic equations for rotational systems.
Angular acceleration is the rate of change of angular velocity over time. It is calculated as the change in angular velocity divided by the change in time. Angular acceleration, like linear acceleration, can be constant or varying. For constant angular acceleration, the angular kinematic equations relating angular displacement, velocity, acceleration, and time can be used. Understanding the relationships between linear and angular quantities like distance, speed, and acceleration is important when analyzing rotational or spinning motion.
Angular momentum is a measure of rotational motion and is equal to the moment of inertia multiplied by angular velocity. It is conserved for rotating objects where no external torque is applied. Many sports techniques and maneuvers rely on changing an object's moment of inertia through motions like tucking or pulling in limbs to increase angular velocity and angular momentum transfer. The Magnus effect causes spinning balls like baseballs to curve due to differences in air pressure above and below the ball, an application of Bernoulli's principle.
The document discusses angles and their measurement in degrees and radians. It defines standard position of an angle, rotation of angles, coterminal angles, reference angles, and the relationship between degrees and radians. Examples are provided for finding measures of angles in standard position, coterminal angles, reference angles, and converting between degrees and radians. Key concepts covered include a full rotation being 360 degrees or 2π radians, and that coterminal angles have the same terminal side.
Trigonometric ratios and functions deal with relationships in triangles, especially right triangles. They can be used both in geometry to solve triangles and analytically by treating trig functions as functions in equations. Trigonometry has been used for over 3000 years, originally to determine lengths and angles in triangles and more recently by treating trig functions as functions in equations. Angles are measured in degrees, minutes, and seconds, and can also be measured in radians where one radian is the central angle that intercepts the same arc length as the radius. Trig functions like sine, cosine, and tangent can be evaluated for any angle using the unit circle or by finding the reference angle in the first quadrant. Trig functions can be graphed by identifying
This document defines angles and angle measure in geometry and trigonometry. It explains that an angle is formed by two rays with a common endpoint, and can be measured in degrees from 0 to 360 degrees. The document discusses angle terminology like initial side, terminal side, standard position, coterminal angles, quadrantal angles, and locating angles by quadrant. It provides examples of finding coterminal angles and sketching angles in standard position. Exercises at the end have the reader practice finding coterminal angles, sketching angles, and determining angle locations.
One complete counterclockwise rotation measures 360° and one complete clockwise rotation measures −360°.
Thus, 1° is equal to 1/360 rotation (counterclockwise) and 90° is equal to 90/360 or 1/4 rotation (counterclockwise).
To find the angle measure in degrees for a given rotation, multiply the amount of rotation by 360° for counterclockwise rotation or by −360° for clockwise rotation.
This document defines and describes different types of angles:
- Acute angles are less than 90 degrees. Obtuse angles are greater than 90 degrees but less than 180 degrees. Right angles are 90 degrees. Straight angles are 180 degrees. Reflex angles are greater than 180 degrees but less than 360 degrees.
- Angles can be calculated based on their relationship to other angles, such as angles around a point adding up to 360 degrees and angles on a straight line adding up to 180 degrees. Vertically opposite angles are always equal.
- When parallel lines are intersected by a transversal, the corresponding angles, alternate interior angles, alternate exterior angles, and interior angles on the same side of the transversal are
This document defines and provides equations for angular distance, velocity, and acceleration. It explains that angular distance is equal to the angular path length. Angular velocity is the rate of change of angular displacement with respect to time. Angular acceleration is the rate of change of angular velocity with respect to time. Linear velocity at a point can be calculated as the sum of angular velocity times the radius of rotation and any linear velocity of the body.
This document provides information on 10 circle theorems including: angles in a semicircle are right angles; opposite angles in a cyclic quadrilateral add up to 180 degrees; equal chords subtend equal angles; and tangents from a point are equal in length. Examples are worked through demonstrating each theorem. Real-life applications are discussed such as using circle theorems and Pythagoras' theorem to calculate distances on Earth and how circle geometry has remained important in theories of atoms and the universe.
Here you can learn all about the math concepts that are hidden in miniature golf. Visit www.putterking.com for more info.
Level 2 - Princess
Area of focus: angles
Topics covered:
> Supplementary angles
> Complementary angles
> Congruent angles
> Adjacent angles
> Linear pairs
> Vertical angles
> Angle bisectors
This document discusses various instruments used for angular measurement including protractors, sine bars, angle gauges, spirit levels, autocollimators, and angle dekkors. It provides details on their construction, principles of operation, types, accuracy, and applications. For example, it explains that a universal bevel protractor can measure angles with an accuracy up to 5 minutes and has a vernier scale to improve readability while a sine bar is used to accurately measure and set angles but is limited to less than 45 degrees.
This section discusses angles and arcs. It defines angles, their measurement in degrees and radians, and how to convert between the two units. It also defines arc length and how to calculate it given the radius and measure of a central angle in radians. Finally, it discusses the relationships between linear speed, angular speed, radius, and time for objects moving in circular motion.
This document covers trigonometric functions including radians, the unit circle, converting between degrees and radians, evaluating trig functions without a calculator, graphing sine, cosine and tangent functions, and writing equations for translated and stretched trig functions. Key points include:
- One radian is the measure of a central angle whose arc length is equal to the radius of the circle.
- Trig functions are evaluated using points on the unit circle or by using special right triangles.
- Sine and cosine graphs have a period of 2π and amplitude of 1, while tangent graphs have a period of π.
- Translated and stretched trig functions can be written by applying transformations to the original equations.
PC_Q2_W1-2_Angles in a Unit Circle Presentation PPTRichieReyes12
This document covers measures of arcs in a unit circle. It discusses angle measure in degrees and radians, how to convert between the two units, and illustrates angles in standard position and coterminal angles. It also explains that in a unit circle, an arc with length 1 intercepts a central angle measuring 1 radian. The length of an arc and area of a sector are directly proportional to the radian measure of the intercepting central angle.
1. Rotational motion refers to the movement of an object around an axis or center point, with each particle moving in a circular path around the axis of rotation.
2. The key difference between circular and rotational motion is that in rotational motion, the object rotates around a fixed axis, while in circular motion the object simply moves in a circle without rotation around an axis.
3. Moment of inertia quantifies an object's resistance to changes in its rotational motion, playing a similar role for rotational motion as mass does for translational motion. It depends on the object's mass distribution and distance from the axis of rotation.
Here are the key points about torque:
- Torque depends on the force applied and the lever arm (distance from the axis of rotation).
- A longer lever arm results in greater torque for the same force.
- Torque is maximized when the force is applied perpendicular to the lever arm.
Therefore, the picture with the greatest torque is C, as it has the longest lever arm (distance from the axis of rotation). Even though the force in A is greater, the shorter lever arm means the torque is less than in C.
This document defines key angle terminology and concepts including: classifying angles as acute, obtuse, right, or quadrantal based on their measure; complementary and supplementary angles; converting between degree-minute-second and decimal degree angle measures; and finding coterminal angles. It provides examples such as calculating the complement and supplement of a 40° angle, adding and subtracting angles in degree-minute-second form, and converting between decimal degrees and degree-minute-second notation.
The document defines various terms related to circle geometry such as radius, diameter, chord, tangent, arc, and angle. It then lists 10 rules of circle geometry pertaining to angles, chords, tangents, and polygons. Finally, it provides some practice questions and links for additional resources on circle geometry concepts.
The document provides a math review covering topics in algebra, geometry, trigonometry, and statistics. It defines concepts like negative numbers, exponents, square roots, order of operations, lines, angles, trigonometric functions, and averages. Formulas are presented for topics like quadratic equations, the Pythagorean theorem, laws of sines and cosines, percentages, and standard deviation. Examples are included to illustrate key ideas.
The document provides an overview of solving biomechanics problems, distinguishing between qualitative and quantitative approaches. It discusses solving formal quantitative problems through a step-by-step process of understanding the problem, identifying known and unknown values, selecting applicable formulas, performing calculations, and checking results. Examples of quantitative problems are provided along with discussions of units, conversions, and evaluating whether answers are reasonable.
1) The document defines three types of motion - translation, angular, and general - and describes anatomical reference positions, planes, axes, and directional terms used to qualitatively analyze human movement.
2) It provides details on planar movements including flexion/extension, abduction/adduction, and internal/external rotation in the sagittal, frontal, and transverse planes respectively.
3) Qualitative analysis of human movement involves descriptive observation of technique and performance to identify causes of problems and differentiate unrelated factors, and the document outlines steps to plan and conduct such an analysis.
The document defines vectors and describes various vector operations that can be performed, including:
- Graphical vector addition using the tip-to-tail method
- Numerical representation of vectors using magnitude, direction, and components
- Resolution of a vector into perpendicular components
- Composition and decomposition of vectors using graphical and numerical methods
- Scalar multiplication and subtraction of vectors
It also provides examples of how to use vectors to solve problems graphically or numerically.
The document defines key concepts in linear kinematics including:
1) Spatial reference frames which provide axes to describe position and direction in 1, 2, or 3 dimensions.
2) Linear concepts such as position, displacement, distance, velocity, and speed. Displacement is the change in position, velocity is the rate of change of position, and speed is the distance traveled per unit time.
3) Methods for calculating displacement, velocity, and speed using change in position over change in time. Velocity is a vector while speed is a scalar.
This document introduces the concepts of linear acceleration, computing acceleration from changes in velocity over time, and the differences between average and instantaneous acceleration. It provides examples of calculating acceleration from velocity data and graphs, and reviews the laws of constant acceleration for relating changes in velocity, displacement, and time when acceleration is constant. Key topics covered include computing acceleration from changes in velocity and time, using graphs of velocity over time to determine acceleration, and applying the kinematic equations for constant acceleration.
The document discusses projectile motion, which describes the trajectory of objects in free fall under only the forces of gravity and air resistance. It defines a projectile and explains how gravity influences the vertical and horizontal components of motion differently. Key factors that determine a projectile's trajectory include the projection angle, speed, and height. Optimal projection conditions exist that maximize distance or height based on these factors. Examples are provided to demonstrate how to calculate values like maximum height, flight time, and distance for a given projectile scenario.
Kinetics is the study of the relationship between forces acting on a system and its motion. It includes concepts like inertia, mass, force, weight, torque, and impulse. Forces can cause both acceleration and deformation of objects. Stress is the force distributed over an area, while pressure is the stress due to compression. Materials respond elastically to small loads but experience permanent plastic deformation above the yield point, with rupture occurring at ultimate failure. Repeated cyclic loading reduces the stress needed to cause material failure compared to a single acute load.
This document defines linear kinetics and describes Newton's three laws of motion. Linear kinetics is the study of the relationship between forces and motion for objects undergoing linear or translational motion. Newton's first law states that an object at rest stays at rest or an object in motion stays in motion with constant velocity unless acted upon by an external force. The second law relates the external force on an object to its mass and acceleration. The third law states that for every action force there is an equal and opposite reaction force. Examples are also given to illustrate applications of the laws.
1) Momentum is defined as the product of an object's mass and velocity. It is a vector quantity that represents the quantity of motion.
2) The principle of conservation of momentum states that if the total external force on a system is zero, the total momentum of the system remains constant.
3) Impulse is defined as the change in momentum of an object due to an applied force over time. According to the impulse-momentum theorem, the impulse applied to an object equals its change in momentum.
Mechanical work is the product of the force applied and the displacement in the direction of the force. Power is the rate of work done over time. There are two types of energy: kinetic energy, which is the energy of motion, and potential energy, which is stored energy due to an object's position or deformation. The total energy of an isolated system remains constant according to the principle of conservation of energy. Friction is a force that opposes motion between two surfaces in contact. There are two types of friction: static and kinetic friction.
1) Angular kinetics is the study of forces that cause rotation or torques. Torque is a measure of how much a force causes an object to rotate and depends on the force magnitude and its moment arm.
2) The moment arm is the distance from the axis of rotation to where the force is applied. Torque is calculated by multiplying the force by the moment arm.
3) Resultant joint torque is the single torque that has the same rotational effect as all the individual torques acting on a joint. It provides a simplified view of which muscle groups are most active at a joint.
1) A free body diagram is used to represent all external forces and torques acting on a system. It is an important step in solving kinetics problems.
2) The document provides guidance on constructing free body diagrams including identifying the system, drawing external forces and torques, and specifying the point of application and direction.
3) Lever systems use an effort force to move a load force. There are three classes of levers that vary based on the relative positions of the effort, load, and fulcrum. Mechanical advantage determines the trade off between force and distance of movement.
Stability and balance refer to an object's ability to resist changes to its equilibrium state and return to its original position if disturbed. Key factors that influence stability include an object's mass/moment of inertia, its base of support, the position of its center of mass relative to the base of support, and surface friction. Static balance requires keeping the center of mass over the base of support, while dynamic balance involves controlling the center of mass during movements to prevent losing equilibrium and falling.
- A system is at static equilibrium when it is at rest and experiences no translation or rotation (according to Newton's 1st law). The net external forces and torques on the system must equal zero.
- Dynamic equilibrium applies to accelerating rigid bodies (according to Newton's 2nd law). The net external forces must equal mass times acceleration, and net torque must equal moment of inertia times angular acceleration.
- Inverse dynamics uses measured joint positions, ground reaction forces, and segment parameters to compute unknown joint forces and torques that produce the observed motion. Segments are analyzed individually from distal to proximal.
Biomechanics is the application of mechanical principles to the study of living organisms like the human body. It has two main sub-branches - statics which looks at systems at rest or in constant motion, and dynamics which examines accelerated systems. Biomechanics is used by professionals in sports, health, rehabilitation and engineering to improve performance, prevent and treat injuries, reduce physical declines, improve mobility, and aid product design. The goal of this introduction is to define key biomechanics concepts and illustrate its wide-ranging applications.
1) Intelligence is defined as the ability to act appropriately in uncertain environments in order to achieve goals and succeed.
2) Natural intelligence evolved through natural selection to produce behaviors that increase survival and reproduction.
3) More intelligent individuals and groups are better able to sense their environment, make decisions, and take actions that provide biological advantages over less intelligent competitors.
1. Angular Kinematics
Objectives:
• Introduce the angular concepts of absolute
and relative angles, displacement, distance,
velocity, and speed
• Learn how to compute angular displacement,
velocity, and speed
• Learn to compute and estimate instantaneous
angular velocity
Angular Kinematics
Kinematics
• The form, pattern, or sequencing of movement
with respect to time
• Forces causing the motion are not considered
Angular Motion (Rotation)
• All points in an object or system move in a circle
about a single axis of rotation. All points move
through the same angle in the same time
Angular Kinematics
• The kinematics of particles, objects, or systems
undergoing angular motion
1
2. Angular Kinematics & Motion
• Most volitional movement is
performed through rotation SHOULDER
of the body segments NECK
• The body is often analyzed
ELBOW
as a collection of rigid, LUMBAR
rotating segments linked at HIP
the joint centers
• This is a rough
KNEE
approximation
ANKLE
Measuring Angles
Degrees: Radians:
90 π/2
57.3° 1 radian
180 0, 360 π 0, 2 π
270 3π/2
1 radian = 57.3° π = 3.14159
1 revolution = 360° = 2π radians Note: Excel uses
θ(degrees) = (180/π)× θ(radians) radians!
2
3. Positive vs. Negative Angles
By convention, when describing angular kinematics:
• positive angles counterclockwise rotation
• negative angles clockwise rotation
Positive: Negative:
+90° -270°
+57.3°
+180° 0,+360° -180° 0,-360°
-57.3°
+270° -90°
Absolute Angle (or Angle of Inclination)
• Angular orientation of a line segment with respect
to a fixed line of reference
• Use the same reference for all absolute angles
θ
θ
Trunk angle Trunk angle
from horizontal from vertical
3
4. Angular Displacement
• Change in the angular position or orientation of a
line segment
• Doesn’t depend on the path between orientations
• Has angular units (e.g. degrees, radians)
axis of rotation
initial final
orientation orientation
angular
displacement
Computing Angular Displacement
• Compute angular displacement (∆θ) by subtraction
of angular positions:
∆θ = θfinal – θinitial
∆θ
final
orientation θfinal initial orientation
θinitial
axis of rotation
4
5. Angular Distance
• Sum of the magnitude of all angular changes
undergone by a rotating body
• Has angular units of length (e.g. degrees, radians)
• Distance ≥ (Magnitude of displacement)
Angular final orientation
Distance = 225°
-90°
intermediate
Angular
orientation 135° Displacement = 45°
axis of rotation initial orientation
Example Problem #1
A figure skater spins 10.5 revolutions in a
clockwise direction, pauses, then spins 60°
counterclockwise before skating away.
What were the total angular distance and angular
displacement?
5
6. Relative Angle
• Angle between two line segments
• Compute relative angle by subtraction of absolute
angular positions:
θ(1→2) = θ2 – θ1
θ(1→2) segment 1
segment 2
θ2
θ1
axis of rotation
Joint Angles
• Joint angles are relative angles between longitudinal
axes of adjacent segments (or between anterior-
posterior axes for internal rotation)
θelbow
θshoulder
θhip
θknee Use a consistent sign
convention for joint angles
θankle (e.g. + = flexion)
6
7. Computing Joint Angles
• Involves subtracting absolute angles of segments
• Exact formula and order of subtraction depends on
the joint and the convention chosen
θknee = θleg– θthigh θknee = 180° + θthigh– θleg
HIP θthigh HIP θthigh
θknee
θleg θleg
KNEE ANKLE KNEE
θknee ANKLE
Angular Velocity
• The rate of change in the absolute or relative angular
position or orientation of a line segment
change in angular angular
angular position displacement
velocity = =
change in time change in time
• Shorthand notation:
θfinal – θinitial ∆θ
ω= =
tfinal – tinitial ∆t
• Has units of (angular units)/time (e.g. radians/s, °/s)
7
8. Angular Speed
• The angular distance traveled divided by the time
taken to cover it
• Equal to the average magnitude of the
instantaneous angular velocity over that time.
angular distance
angular speed =
change in time
• Has units of (angular units)/time (e.g. radians/s, °/s)
Angular Speed vs. Velocity
Angular end of follow-
Distance = 225° through
-90°
end of
Angular
backswing 135° Displacement = 45°
tennis player racquet at start
Assume tennis stroke shown takes 0.75 s:
225° +45°
Speed = Velocity =
0.75 s 0.75 s
= 300°/s = +60°/s
8
9. Example Problem #2
The figure skater of Problem #1 completes the
first (clockwise) spin in 3 s, pauses for 1 s,
then completes the second (counterclockwise)
spin in 0.3 s.
What were her average angular velocity and
average angular speed during the first spin?
What were her average angular velocity and
average angular speed for the skill as a
whole?
Example Problem #3
A person is performing a squat exercise. She
starts from a standing (i.e. anatomical) position.
At her lowest point, 2 seconds later, her knees are
flexed to 60° and her hips are flexed to 90° from
the anatomical position.
1 second later, she has risen back to the standing
position and completed the exercise.
What were the average knee and hip angular
velocities during each phase of the exercise?
for the exercise as a whole?
9
10. Average vs. Instantaneous Velocity
• Previous formulas give us the average velocity
between an initial time (t1) and a final time (t2)
• Instantaneous angular velocity is the angular
velocity at a single instant in time
• Can estimate instantaneous angular velocity using
the central difference method:
θ (at t1 + ∆t) – θ (at t1 – ∆t)
ω (at t1) =
2 ∆t
where ∆t is a very small change in time
Angular Velocity as a Slope
• Graph of angular position vs. time
slope = instantaneous
ω at t1
θ (degrees)
slope = average
ω from t1 to t2
∆θ(1→2)
∆t(1→2)
∆t
t1 t2 time (s)
10
11. Estimating Angular Velocity
Identify points with
zero slope = points
θ (deg)
with zero velocity
Portions of the curve
with positive slope
time (s) have positive velocity
(i.e. velocity in the
ω (deg/s)
+ direction)
Portions of the curve
with negative slope
0
time (s) have negative velocity
(i.e. velocity in the
– direction)
Example Problem #4
A gymnast swings back and forth from the high
bar as shown below. Sketch her angular
velocity.
80
60
40
angle (deg)
20
0
0 2 4 6 8 10
-20
θ
-40
-60
time (s)
11