Chapter 8

Chapter 8

Rotational Kinematics

8.1 Rotational Motion and Angular Displacement

In the simplest kind of rotation,points on a rigid object move oncircular paths around an axis ofrotation.


The angle through which the object rotates is called theangular displacement.

o


DEFINITION OF ANGULAR DISPLACEMENT

•When a rigid body rotates about a fixed axis, the angular displacement is the angle swept out by a line passing through any point on the body and intersecting the axis of rotation perpendicularly.

•By convention, the angular displacement is positive if it is counterclockwise and negative if it is clockwise.

•SI Unit of Angular Displacement: radian (rad)


r

s

Radius

length Arcradians)(in

For a full revolution:

360rad 2 rad 22

r

r


Example 1 Adjacent Synchronous Satellites

Synchronous satellites are put into an orbit whose radius is 4.23×107m.

If the angular separation of the twosatellites is 2.00 degrees, find the arc length that separates them.


rad 0349.0deg360

rad 2deg00.2

miles) (920 m1048.1

rad 0349.0m1023.46

7

rs

r

s

Radius

length Arcradians)(in

8.2 Angular Velocity and Angular Acceleration

o

How do we describe the rate at which the angular displacement is changing?


DEFINITION OF AVERAGE ANGULAR VELOCITY

timeElapsed

ntdisplacemeAngular locity angular ve Average

ttt o

o

SI Unit of Angular Velocity: radian per second (rad/s)


Example 2 Gymnast on a High Bar

A gymnast on a high bar swings throughtwo revolutions in a time of 1.90 s.

Find the average angular velocityof the gymnast.


rad 6.12rev 1

rad 2rev 00.2

srad63.6s 90.1

rad 6.12


Changing angular velocity means that an angular acceleration is occurring.

DEFINITION OF AVERAGE ANGULAR ACCELERATION

ttt o

o

timeElapsed

locityangular vein Change on acceleratiangular Average

SI Unit of Angular acceleration: radian per second squared (rad/s2)


Example 3 A Jet Revving Its Engines

As seen from the front of the engine, the fan blades are rotating with an angular speed of -110 rad/s. As theplane takes off, the angularvelocity of the blades reaches-330 rad/s in a time of 14 s.

Find the angular acceleration, assuming it tobe constant.


2srad16s 14

srad110srad330

8.3 The Equations of Rotational Kinematics

atvv o

tvvx o 21

axvv o 222

221 attvx o

Five kinematic variables:

1. displacement, x

2. acceleration (constant), a

3. final velocity (at time t), v

4. initial velocity, vo

5. elapsed time, t

Recall the equations of kinematics for constant acceleration.


to

to 21

222 o

221 tto

The equations of rotational kinematics for constant angular acceleration:

ANGULAR DISPLACEMENT

ANGULAR VELOCITY

ANGULAR ACCELERATION

TIME


Reasoning Strategy1. Make a drawing.

2. Decide which directions are to be called positive (+) and negative (-).

3. Write down the values that are given for any of the five kinematic variables.

4. Verify that the information contains values for at least three of the five kinematic variables. Select the appropriate equation.

5. When the motion is divided into segments, remember that the final angular velocity of one segment is the initial velocity for the next.

6. Keep in mind that there may be two possible answers to a kinematics problem.


Example 4 Blending with a Blender

The blades are whirling with an angular velocity of +375 rad/s whenthe “puree” button is pushed in.

When the “blend” button is pushed,the blades accelerate and reach agreater angular velocity after the blades have rotated through anangular displacement of +44.0 rad.

The angular acceleration has a constant value of +1740 rad/s2.

Find the final angular velocity of the blades.


θ α ω ωo t

+44.0 rad +1740 rad/s2 ? +375 rad/s

222 o

srad542rad0.44srad17402srad375

2

22

2

o

8.4 Angular Variables and Tangential Variables

velocityl tangentiaTv

speed l tangentiaTv


tr

t

r

t

svT

t

rad/s)in ( rvT


t

rt

rr

t

vva ooToT

T

to

)rad/sin ( 2raT


Example 5 A Helicopter Blade

A helicopter blade has an angular speed of 6.50 rev/s and anangular acceleration of 1.30 rev/s2.For point 1 on the blade, findthe magnitude of (a) thetangential speed and (b) thetangential acceleration.


srad 8.40rev 1

rad 2

s

rev 50.6

sm122srad8.40m 3.00 rvT

a. Calculating angular speed and tangential speed.


22 sm5.24srad17.8m 3.00 raT

22

srad 17.8rev 1

rad 2

s

rev 30.1

b. Calculating angular acceleration and tangential acceleration.

8.5 Centripetal Acceleration and Tangential Acceleration

rad/s)in ( 2

22

r

r

r

r

va T

c


Example 6 A Discus Thrower

Starting from rest, the thrower accelerates the discus to a final angular speed of +15.0 rad/s in a time of 0.270 s before releasing it. During the acceleration, the discus moves in a circular arc of radius 0.810 m.

Find the magnitude of the total acceleration.


2

22

sm182

srad0.15m 810.0

rac

2sm0.45

s 0.270

srad0.15m 810.0

t

ω-ωrra o

T

22222 sm187sm0.45sm182 cT aaa

With use of the Pythagorean Theorem, we can determine the overall magnitude of the acceleration.

8.6 Rolling Motion

The tangential speed of a point on the outer edge of the tire is equal to the speed of the car over the ground.

v = rThis may seem confusing, but there are

two ways to explain.

1. Since the arc length and distance along the ground are equal

s/t = d/t = v.

2. Secondly, the speed of any point on the tire is relative to the axle, not the road. A point at the top of rotation will be moving at twice the velocity relative to the road while a point on the tire that makes contact with the road has a zero component of linear velocity. Hence, the average velocity of any point on the

tire will be v.

8.6 Rolling Motion(cont.)

The acceleration of the vehicle can be analyzed in a similar fashion to linear and tangential velocity.

Thus:

ra

8.6 Rolling Motion

Example 7 An Accelerating Car

Starting from rest, the car accelerates for 20.0 s with a constant linear acceleration of 0.800 m/s2. The radius of the tires is 0.330 m.

What is the angle through which each wheel has rotated?

8.6 Rolling Motion

221 tto

θ α ω ωo t? -2.42 rad/s2 0 rad/s 20.0 s

22

srad42.2m 0.330

sm800.0

r

a

rad 484s 0.20srad42.2 22221

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