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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)
r
s
Radius
length Arcradians)(in
For a full revolution:
360rad 2
rad 22
r
r
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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 0349.0deg360
rad 2deg00.2
miles) (920 m1048.1
rad 0349.0m1023.46
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rs
r
s
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.
r
s
Radius
length Arcradians)(in
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Angular Displacement
o timeElapsed
ntdisplacemeAngular 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 1
rad 2rev 00.2
srad63.6s 90.1
rad 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.
timeElapsed
locityangular 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 Kinematics
Example 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
rt
r
t
svT
t
rad/s)in ( rvT
t
rt
rr
t
vva ooToTT
)rad/sin ( 2raT
to
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Angular & Tangential Variables
Example 6 A Helicopter Blade
A 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 1
rad 2
s
rev 50.6
sm122srad8.40m 3.00 rvT
22 sm5.24srad17.8m 3.00 raT
22
srad 17.8rev 1
rad 2
s
rev 30.1
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Centripetal & Tangential Acceleration
222
r
r
r
r
va Tc
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
t
ω-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/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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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.