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Ch 8. Rotational Kinematics

Rotation points move on circular paths around an axis of rotation

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Rotational Motion & Angular Displacement

Angular displacement angle through which the object rotates

o

Units = Radian (rad)

rs

Radius

length Arcradians)(in

For a full revolution:

360rad 2

rad 22 rr

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Rotational Motion & Angular Displacement

Example: 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 0349.0deg360rad 2deg00.2

miles) (920 m1048.1

rad 0349.0m1023.46

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rs

rs

Radius

length Arcradians)(in

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8.1 Rotational Motion and Angular Displacement

Example: Total Solar Eclipse

Diameter of the sun = ~400 times that of the moon. The sun is also ~400 times farther from the earth than moon.

Compare the angle subtended by the moon to the angle subtended by the sun and explain why this leads to a total solar eclipse.

rs

Radius

length Arcradians)(in

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Angular Displacement

o timeElapsedntdisplacemeAngular locity angular ve Average

ttt o

o

Units : radian per second (rad/s)

ttt

00

limlim

Instantaneous Angular Velocity

Average Angular Velocity

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Angular Velocity & Angular Acceleration

Example 3 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 1rad 2rev 00.2

srad63.6s 90.1rad 6.12

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Angular Acceleration

ttt o

o

on acceleratiangular Average

Units: radian per second squared (rad/s2)

Example 4 A Jet Revving Its Engines

A jet’s fan blades are rotating with an angular speed of -110 rad/s. As the plane takes off, the angular velocity of the blades reaches -330 rad/s in a time of 14 s. Find the angular acceleration, assuming it to be constant.

timeElapsedlocityangular vein Change

s 14

srad110srad330

2srad16

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Linear Kinematics

atvv o

tvvx o 21

axvv o 222

221 attvx o

Five kinematic variables:

1. displacement, x2. acceleration (constant), a3. final velocity (at time t), v4. initial velocity, vo

5. elapsed time, t

Rotational Kinematics

to

to 21

222 o

221 tto

ANGULAR VELOCITY

ANGULAR ACCELERATION

ANGULAR DISPLACEMENT TIME

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8.3 Equations of Rotational Kinematics Reasoning Strategy1. Make a drawing.

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

3. Write down the values 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.

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Rotational KinematicsExample 5 Blending with a Blender

The blades are whirling with angular velocity of +375 rad/s. The blades accelerate and reach a greater angular velocity after the blades have rotated through an angular 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

22 o

rad0.44srad17402srad375 22

srad542

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Angular Variables & Tangential Variables

velocityl tangentiaTv

speed l tangentiaTv

t

rtr

tsvT

t

rad/s)in ( rvT

t

rtrr

tvva ooToT

T

)rad/sin ( 2raT

to

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Angular & Tangential VariablesExample 6 A Helicopter BladeA helicopter blade has an angular speed of 6.50 rev/s and an angular acceleration of 1.30 rev/s2. For point 1 on the blade, find the magnitude of (a) the tangential speed and (b) the tangential acceleration.

srad 8.40rev 1rad 2

srev 50.6

sm122srad8.40m 3.00 rvT

22 sm5.24srad17.8m 3.00 raT

22 srad 17.8

rev 1rad 2

srev 30.1

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Centripetal & Tangential Acceleration

222

rrr

rva T

c

rad/s)in (

Example 7 A Discus Thrower

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

222 sm182srad0.15m 810.0 rac

2sm0.45s 0.270

srad0.15m 810.0

tω-ωrra o

T

22222 sm187sm0.45sm182 cT aaa

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Rolling Motion

rv

Tangential speed of outer edge of tire = speed of car over ground.

ra Example 8 An Accelerating Car

Starting from rest, a car accelerates for 20 s with acceleration of 0.800 m/s2. Radius of tires is 0.330 m. What is the angle through which each wheel has rotated?

221 tto

θ α ω ωo t? -2.42

rad/s20

rad/s20.0

s

22

srad42.2m 0.330

sm800.0

ra

rad 484s 0.20srad42.2 22221

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8.7 The Vector Nature of Angular Variables

Right-Hand Rule: Grasp the axis of rotation with your right hand, so that your fingers circle the axisin the same sense as the rotation.

Your extended thumb points along the axis in thedirection of the angular velocity.