Chapter 11A Chapter 11A –– Angular MotionAngular Motion

A PowerPoint Presentation by

Paul E. Tippens, Professor of Physics

Southern Polytechnic State University

A PowerPoint Presentation byA PowerPoint Presentation by

Paul E. Tippens, Professor of PhysicsPaul E. Tippens, Professor of Physics

Southern Polytechnic State UniversitySouthern 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 rotationalDraw analogies relating rotational--motion motion parameters (parameters (, , , , ) to linear () to linear (x, v, ax, 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 power1. Rotational work, energy, and power

2. Rotational kinetic energy and 2. Rotational kinetic energy and momentummomentum

3. Conservation of angular momentum3. 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 rev1 rev == 360 360 00 = 2= 2

radrad

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 radianradian..

Definition of the RadianDefinition of the Radian

One One radian 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 ss equal to the radius equal to the radius RR of the circle.of the circle.

1 rad = = 57.30R

R

ssR

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 cm50 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 m20 m??

h = h = 20 m20 m

RR = 40 rad= 40 rad

Now, Now, 1 rev = 21 rev = 2

radrad

= 6.37 rev = 6.37 rev

1 rev40 rad2 rad

20 m0.50 m

sR

Example 2:Example 2: A bicycle tire has a radius of A bicycle tire has a radius of 25 cm25 cm. If the wheel makes . If the wheel makes 400 rev400 rev, how , how far will the bike have traveled?far will the bike have traveled?

= 2513 rad= 2513 rad

s = s = R = R = 25132513 rad (0.25 m)rad (0.25 m)

s = 628 ms = 628 m

2 rad400 rev1 rev

Angular VelocityAngular VelocityAngular velocityAngular velocity,,

is the rate of change in is the rate of change in

angular displacement. (radians per second.)angular displacement. (radians per second.)

f Angular frequency f (rev/s). ff Angular frequency Angular frequency ff (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 f (rev/s or rpm):(rev/s or rpm):

Angular velocity in rad/s.Angular velocity in rad/s.

tt

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 cm20 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 m10 m in in 5 s5 s??

h = h = 10 m10 m

R

= 10.0 rad/s = 10.0 rad/s

t

50 rad

5 s

= 50 rad= 50 rad10 m

0.20 msR

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 mh = 10 m

R

f = 95.5 rpmf = 95.5 rpm

2 or 2

f f

10.0 rad/s 1.59 rev/s2 rad/rev

f

Or, since 60 s = 1 min:Or, since 60 s = 1 min:

rev 60 s rev1.59 95.51 min min

fs

Angular AccelerationAngular AccelerationAngular accelerationAngular 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 ( ) 2f Since ft

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 1616 rad/srad/s after a time of after a time of 4 s4 s. What is the . What is the average angular acceleration?average angular acceleration?

h = h = 20 m20 m

R

= 4.00 rad/s2 = 4.00 rad/s2

0

f o fort t

2

16 rad/s rad4.004 s s

Angular and Linear SpeedAngular and Linear Speed

From the definition of angular displacement:

s = R Linear vs. angular displacement

v = Rs Rv Rt t t

Linear speed = angular speed x radiusLinear 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 = Rv v R vv Rt t t

Linear accel. = angular accel. x radiusLinear 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); vvAfAf = 4= 4 m/sm/s

vAf = Bf R1 = (20 rad/s)(0.4 m); vvBfBf = 8= 8 m/sm/s

Acceleration ExampleAcceleration Example

R1 = 20 cm R2 = 40 cm

What is the What is the averageaverage 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/s2= 5.00 rad/s2

a = R = (5 rad/s2)(0.4 m) a = 2.00 m/s2a = 2.00 m/s2

0 20 rad/s4 s

f

t

Angular vs. Linear ParametersAngular vs. Linear Parameters

Angular accelerationAngular acceleration is the time is the time rate of change in angular velocity.rate of change in angular velocity.

0f

t

Recall the definition of Recall the definition of linear linear acceleration acceleration aa from kinematicsfrom kinematics..

0fv va

t

But, But, aa = = RR and and vv = = RR, so that we may write:, so that we may write:

0fv va

t

becomes 0fR RR

t

A Comparison: Linear vs. AngularA Comparison: Linear vs. Angular

0

2fv v

s vt t

0

2ft t

f o t f ov v at

210 2t t 21

0 2s v t at 21

2f t t

2 202 f 2 2

02 fas v v

212fs v t at

Linear Example:Linear Example: A car traveling initially A car traveling initially at at 20 m/s20 m/s comes to a stop in a distance comes to a stop in a distance of of 100 m100 m. What was the acceleration?. What was the acceleration?

100 m100 m

vvoo = 20 m/s= 20 m/s vvff = 0 m/s= 0 m/s

Select Equation:2 2

02 fas v v

a = = 0 - vo

2

2s

-(20 m/s)2

2(100 m) a = -2.00 m/s2a = -2.00 m/s2

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

02 f

= = 0 - o

2

2

-(62.8 rad/s)2

2(314 rad) = -6.29 m/s2 = -6.29 m/s2

Ro = 600 rpm

f = 0 rpm

= 50 rev

2 rad 1 min600 62.8 rad/smin 1 rev 60 srev

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 rpm100 rpm and undergoes a constant and undergoes a constant counterclockwise acceleration of counterclockwise acceleration of 3 rad/s3 rad/s22 for for 2 s2 s. What is the angular displacement?. What is the angular displacement?

= -14.9 rad = -14.9 rad

Given:Given: o = -100 rpm; t = 2 s

= +2 rad/s2

2 21 12 2( 10.5)(2) (3)(2)ot t

rev 1 min 2 rad100 10.5 rad/smin 60 s 1 rev

= -20.9 rad + 6 rad

Net displacement is clockwise (-)

R

Summary of Formulas for RotationSummary of Formulas for Rotation

0

2fv v

s vt t

0

2ft t

f o t f ov v at

210 2t t 21

0 2s v t at 21

2f t t

2 202 f 2 2

02 fas v v

212fs v t at

CONCLUSION: Chapter 11A CONCLUSION: Chapter 11A Angular MotionAngular Motion