Angular MotionAngular Motion

Objectives:Objectives:

• Define and apply concepts of angular Define and apply concepts of angular displacement, velocity, and displacement, velocity, and acceleration.acceleration.

• Draw analogies relating rotational-Draw analogies relating rotational-motion parameters (motion 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.

Rotational Displacement, Rotational Displacement,

Consider a disk that rotates from A to Consider a disk that rotates from A to B:B:

A

B

Angular displacement Angular displacement ::

Measured in Measured in revolutions, degrees, or revolutions, degrees, or

radians.radians.

1 rev1 rev == 360 360 00 = 2= 2 radrad

The best measure for rotation of The best measure for rotation of rigid bodies is the rigid bodies is the radianradian..

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 RadianOne One radian radian is the angle is the angle subtended at the center of a circle subtended at the center of a circle by an arc length by an arc length ss equal to the equal to the radius radius RR of the circle. of the circle.

1 rad = = 57.30

R

R

ss

R

s

R

Example 1:Example 1: A rope is wrapped many A rope is wrapped many times around a drum of radius times around a drum of radius 50 cm50 cm. . How many revolutions of the drum are How many revolutions of the drum are required to raise a bucket to a height required to raise a bucket to a height of 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 rad

2 rad

20 m

0.50 m

s

R

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

1 rev

Angular VelocityAngular VelocityAngular velocityAngular velocity,, is the rate of is the rate of change in angular displacement. change in angular displacement. (radians per second.)(radians per second.)

ff Angular frequency Angular frequency ff (rev/s).(rev/s).ff Angular frequency Angular frequency ff (rev/s).(rev/s).

Angular velocity can also be given as Angular velocity can also be given as the frequency of revolution, the frequency of revolution, f f (rev/s or (rev/s or rpm):rpm):

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

tt

Example 3:Example 3: A rope is wrapped many A rope is wrapped many times around a drum of radius times around a drum of radius 20 cm20 cm. . What is the angular velocity of the What is the angular velocity of the drum if it lifts the bucket to drum if it lifts the bucket to 10 m10 m in in 5 5 ss??

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 m

s

R

Example 3:Example 3: In the previous example, In the previous example, what is the frequency of revolution for what is the frequency of revolution for the drum? Recall that the drum? 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/s1.59 rev/s

2 rad/revf

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

rev 60 s rev1.59 95.5

1 min minf

s

Angular AccelerationAngular AccelerationAngular accelerationAngular acceleration is the rate of is the rate of

change in angular velocity. (Radians per change in angular velocity. (Radians per sec per sec.)sec per sec.)

The angular acceleration can also be The angular acceleration can also be found from the change in frequency, as found from the change in frequency, as follows:follows:

2 ( ) 2

fSince f

t

2 ( ) 2

fSince f

t

2 Angular acceleration (rad/s )t

2 Angular acceleration (rad/s )

t

Example 4:Example 4: The block is lifted from The block is lifted from rest until the angular velocity of the rest until the angular velocity of the drum is drum is 1616 rad/srad/s after a time of after a time of 4 s4 s. . What is the average angular What is the average angular acceleration?acceleration?

h = h = 20 m20 m

R

= 4.00 rad/s2 = 4.00 rad/s2

0

f o fort t

2

16 rad/s rad4.00

4 s s

Angular and Linear SpeedAngular and Linear SpeedFrom the definition of angular

displacement:s = R Linear vs. angular

displacement

v = Rs R

v Rt t t

Linear speed = angular speed x radius

Linear speed = angular speed x radius

Angular and Linear Angular and Linear Acceleration:Acceleration:

From the velocity relationship we have:

v = R Linear vs. angular velocity

a = Rv v R v

v Rt t 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 Consider flat rotating disk: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 and linear angular and linear acceleration at B?acceleration at B?

R1

R2

A

B = 0; f = 20 rad/s t = 4 s

Consider flat rotating Consider flat rotating disk: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/s

4 sf

t

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

f o t f o t f ov v at f ov v at

210 2t t 21

0 2t t 210 2s v t at 21

0 2s v t at

212f t t 21

2f t t

2 202 f 2 2

02 f 2 202 fas v v 2 2

02 fas v v

212fs v t at 21

2fs v t at

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

100 100 mm

vvoo = 20 = 20 m/sm/s

vvff = 0 = 0 m/sm/s

Select Equation:

2 202 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:

2 202 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/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 5:Example 5: A drum is rotating clockwise A drum is rotating clockwise initially at initially at 100 rpm100 rpm and undergoes a and undergoes a constant counterclockwise acceleration constant counterclockwise acceleration of of 3 rad/s3 rad/s22 for for 2 s2 s. What is the angular . What is the angular displacement?displacement?

= -14.9 rad = -14.9 rad

Given:Given: o = -100 rpm; t = 2 s = +3 rad/s2

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

rev 1 min 2 rad100 10.5 rad/s

min 60 s 1 rev

= -20.9 rad + 6 rad

Net displacement is clockwise (-)

R

Summary of Formulas for RotationSummary of Formulas for Rotation

f o t f o t f ov v at f ov v at

210 2t t 21

0 2t t 210 2s v t at 21

0 2s v t at

212f t t 21

2f t t

2 202 f 2 2

02 f 2 202 fas v v 2 2

02 fas v v

212fs v t at 21

2fs v t at