Chapter 11A Chapter 11A –Angular Motion Angular Motion A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University A PowerPoint Presentation by A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Paul E. Tippens, Professor of Physics Southern Polytechnic State University Southern Polytechnic State University © 2007
WIND TURBINES such as these can generate significant energy in a way that is environ- mentally friendly and renewable. The concepts of rotational acceleration, angular velocity, angular displacement, rotational inertia, and other topics discussed in this chapter are useful in describing the operation of wind turbines.
Objectives: After completing this Objectives: After completing this module, you should be able to: module, you should be able to: • • Define and apply concepts of angular Define and apply concepts of angular displacement, velocity, and acceleration. displacement, velocity, and acceleration. •Draw analogies relating rotational Draw analogies relating rotational -motion parameters ( parameters ( , , ) to linear ( ) to linear (x, v, a x, v, a) and solve rotational problems. and solve rotational problems. • • Write and apply relationships between Write and apply relationships between linear and angular parameters. linear and angular parameters.
Objectives: (Continued) Objectives: (Continued) •Define moment of inertia and apply it for Define moment of inertia and apply it for several regular objects in rotation. several regular objects in rotation. • • Apply the following concepts to rotation: Apply the following concepts to rotation: 1. Rotational work, energy, and power 1. Rotational work, energy, and power 2. Rotational kinetic energy and 2. Rotational kinetic energy and momentum momentum 3. Conservation of angular momentum 3. Conservation of angular momentum
Rotational Displacement, Rotational Displacement, Consider a disk that rotates from A to B: Consider a disk that rotates from A to B: A B Angular displacement Angular displacement : Measured in revolutions, Measured in revolutions, degrees, or radians. degrees, or radians. 1 rev= = 360 0= 2 rad The best measure for rotation of rigid bodies is the radian. The best measure for rotation of The best measure for rotation of rigid bodies is the rigid bodies is the radian.
Definition of the Radian Definition of the Radian One radian is the angle is the angle subtended at subtended at the center of a circle by an arc length the center of a circle by an arc length s equal to the radius equal to the radius R of the circle. of the circle. 1 rad = = 57.3 0 R R s s R
Example 1: Example 1: A rope is wrapped many times A rope is wrapped many times around a drum of radius around a drum of radius 50 cm. How many . How many revolutions of the drum are required to revolutions of the drum are required to raise a bucket to a height of raise a bucket to a height of 20 m?h = 20 m = 40 rad = 40 rad R Now, 1 rev = 2 1 rev = 2 rad = 6.37 rev = 6.37 rev 1 rev 40 rad 2 rad 20 m 0.50 m sR
Example 2: Example 2: A bicycle tire has a radius of A bicycle tire has a radius of 25 cm. If the wheel makes . If the wheel makes 400 rev 400 rev, how far will the bike have traveled? far will the bike have traveled? = 2513 rad = 2513 rad s = R = 2513rad (0.25 m) rad (0.25 m) s = s = 628 m 628 m 2 rad 400 rev 1 rev
Angular Velocity Angular Velocity Angular velocity Angular velocity, is the rate of change in is the rate of change in angular displacement. (radians per second.) angular displacement. (radians per second.) ff Angular frequency f (rev/s). f Angular frequency Angular frequency f (rev/s). (rev/s). Angular velocity can also be given as the Angular velocity can also be given as the frequency of revolution, frequency of revolution, f (rev/s or rpm): (rev/s or rpm): Angular velocity in rad/s. Angular velocity in rad/s. t
Example 3: Example 3: A rope is wrapped many times A rope is wrapped many times around a drum of radius around a drum of radius 20 cm. What is . What is the angular velocity of the drum if it lifts the the angular velocity of the drum if it lifts the bucket to bucket to 10 min 5 s? h = 10 m R = 10.0 rad/s = 10.0 rad/s t 50 rad 5 s = 50 rad = 50 rad 10 m 0.20 m sR
Example 4: Example 4: In the previous example, what In the previous example, what is the frequency of revolution for the drum? is the frequency of revolution for the drum? Recall that Recall that = 10.0 rad/s = 10.0 rad/s. h = 10 m h = 10 m R ff = 95.5 rpm = 95.5 rpm 2 or 2 f f 10.0 rad/s 1.59 rev/s 2 rad/rev f Or, since 60 s = 1 min: Or, since 60 s = 1 min: rev 60 s rev 1.59 95.5 1 min min f s
Angular Acceleration Angular Acceleration Angular acceleration Angular acceleration is the rate of change in is the rate of change in angular velocity. (Radians per sec per sec.) angular velocity. (Radians per sec per sec.) The angular acceleration can also be found The angular acceleration can also be found from the change in frequency, as follows: from the change in frequency, as follows: 2( ) 2 f Since f t 2 Angular acceleration (rad/s ) t
Example 5: Example 5: The block is lifted from rest The block is lifted from rest until the angular velocity of the drum is until the angular velocity of the drum is 16rad/safter a time of after a time of 4 s. What is the . What is the average angular acceleration? average angular acceleration? h = 20 m R = 4.00 rad/s = 4.00 rad/s 22 0 f o f or t t 2 16 rad/s rad 4.00 4 s s
Angular and Linear Speed Angular and Linear Speed From the definition of angular displacement: s = R Linear vs. angular displacement v = R s R v R tt t Linear speed = angular speed x radius Linear speed = angular speed x radius
Angular and Linear Acceleration: Angular and Linear Acceleration: From the velocity relationship we have: v = R Linear vs. angular velocity a = R v vR v v R tt t Linear accel. = angular accel. x radius Linear accel. = angular accel. x radius
Examples: Examples: R1 = 20 cm R2 = 40 cm R1 R2 A B = 0; f = 20 rad/s t = 4 s What is final linear speed at points A and B? Consider flat rotating disk: Consider flat rotating disk: vAf = Af R1 = (20 rad/s)(0.2 m); vAf= 4m/s vAf = Bf R1 = (20 rad/s)(0.4 m); vBf= 8m/s
Acceleration Example Acceleration Example R1 = 20 cm R2 = 40 cm What is the What is the average average angular angular and linear acceleration at B? and linear acceleration at B? R1 R2 A B = 0; f = 20 rad/s t = 4 s Consider flat rotating disk: Consider flat rotating disk: = 5.00 rad/s = 5.00 rad/s 2 2 a = R = (5 rad/s 2)(0.4 m) a = 2.00 m/s a = 2.00 m/s 22 0 20 rad/s 4 s f t
Angular vs. Linear Parameters Angular vs. Linear Parameters Angular acceleration Angular acceleration is the time is the time rate of change in angular velocity. rate of change in angular velocity. f 0 t Recall the definition of Recall the definition of linear acceleration acceleration afrom kinematics from kinematics . f 0 v v a t But, a a = R R and v = R, so that we may write: , so that we may write: f 0 v v a t becomes R R f 0 R t
A Comparison: Linear vs. Angular A Comparison: Linear vs. Angular 0 2 f v v s vt t 0 2 f t t f o t f o v v at 1 2 0 2 t t 1 2 0 2 s v t at 1 2 f 2 t t 2 2 2 f 0 2 2 2as v v f 0 1 2 f 2 s v t at
Linear Example: Linear Example: A car traveling initially A car traveling initially at 20 m/scomes to a stop in a distance comes to a stop in a distance of 100 m. What was the acceleration? . What was the acceleration? 100 m vo= 20 m/s = 20 m/s vf= 0 m/s = 0 m/s Select Equation: 2 2 2as v v f 0 a = = 0 - v o 2 2s -(20 m/s) 2 2(100 m) a = -2.00 m/s a = -2.00 m/s 22
Angular analogy: Angular analogy: A disk (R = 50 cm), rotating at 600 rev/min comes to a stop after making 50 rev. What is the acceleration? Select Equation: 2 2 2 f 0 = = 0 - o 2 2 -(62.8 rad/s) 2 2(314 rad) = -6.29 m/s = -6.29 m/s 2 2 R o = 600 rpm f = 0 rpm = 50 rev 2 rad 1 min 600 62.8 rad/s min 1 rev 60 s rev 50 rev = 314 rad
Problem Solving Strategy: Problem Solving Strategy: Draw and label sketch of problem. Draw and label sketch of problem. Indicate Indicate +direction of rotation. direction of rotation. List givens and state what is to be found. List givens and state what is to be found. Given: ____, _____, _____ ( , , f , ,t ) Find: ____, _____ Select equation containing one and not the other of the unknown quantities, and solve for the unknown.
Example 6: Example 6: A drum is rotating clockwise A drum is rotating clockwise initially at initially at 100 rpm 100 rpm and undergoes a constant and undergoes a constant counterclockwise acceleration of counterclockwise acceleration of 3 rad/s 3 rad/s2for 2 s. What is the angular displacement? . What is the angular displacement? = -14.9 rad = -14.9 rad Given: o = -100 rpm; t = 2 s = +2 rad/s2 1 1 2 2 2 2 ( 10.5)(2) (3)(2) o t t rev 1 min 2 rad 100 10.5 rad/s min 60 s 1 rev = -20.9 rad + 6 rad Net displacement is clockwise (-) R
Summary of Formulas for Rotation Summary of Formulas for Rotation 0 2 f v v s vt t 0 2 f t t f o t f o v v at 1 2 0 2 t t 1 2 0 2 s v t at 1 2 f 2 t t 2 2 2 f 0 2 2 2as v v f 0 1 2 f 2 s v t at
CONCLUSION: Chapter 11A CONCLUSION: Chapter 11A Angular Motion Angular Motion