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Chapter 8. Rotational Kinematics. 8.1 Rotational Motion and Angular Displacement. In the simplest kind of rotation, points on a rigid object move on circular paths around an axis of rotation. 8.1 Rotational Motion and Angular Displacement. The angle through which the - PowerPoint PPT Presentation
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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.
8.1 Rotational Motion and Angular Displacement
The angle through which the object rotates is called theangular displacement.
o
8.1 Rotational Motion and Angular Displacement
DEFINITION OF ANGULAR DISPLACEMENT
When a rigid body rotates about a fixed axis, the angulardisplacement is the angle swept out by a line passing throughany point on the body and intersecting the axis of rotation perpendicularly.
By convention, the angular displacementis positive if it is counterclockwise andnegative if it is clockwise.
SI Unit of Angular Displacement: radian (rad)
8.1 Rotational Motion and Angular Displacement
r
s
Radius
length Arcradians)(in
For a full revolution:
360rad 2 rad 22
r
r
8.1 Rotational Motion and Angular Displacement
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.
8.1 Rotational Motion and Angular Displacement
rad 0349.0deg360
rad 2deg00.2
miles) (920 m1048.1
rad 0349.0m1023.46
7
rs
r
s
Radius
length Arcradians)(in
8.1 Rotational Motion and Angular Displacement
Conceptual Example 2 A Total Eclipse of the Sun
The diameter of the sun is about 400 times greater than that of the moon. By coincidence, the sun is also about 400 times farther from the earth than is the moon.
For an observer on the earth, compare the angle subtendedby the moon to the angle subtended by the sun and explainwhy this result leads to a total solar eclipse.
8.1 Rotational Motion and Angular Displacement
r
s
Radius
length Arcradians)(in
8.2 Angular Velocity and Angular Acceleration
o
How do we describe the rateat which the angular displacementis changing?
8.2 Angular Velocity and Angular Acceleration
DEFINITION OF AVERAGE ANGULAR VELOCITY
timeElapsed
ntdisplacemeAngular locity angular ve Average
ttt o
o
SI Unit of Angular Velocity: radian per second (rad/s)
8.2 Angular Velocity and 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.
8.2 Angular Velocity and Angular Acceleration
rad 6.12rev 1
rad 2rev 00.2
srad63.6s 90.1
rad 6.12
8.2 Angular Velocity and Angular Acceleration
ttt
00
limlim
INSTANTANEOUS ANGULAR VELOCITY
8.2 Angular Velocity and Angular Acceleration
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)
8.2 Angular Velocity and Angular Acceleration
Example 4 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.
8.2 Angular Velocity and Angular Acceleration
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.
8.3 The Equations of Rotational Kinematics
to
to 21
222 o
221 tto
The equations of rotational kinematics for constant angular acceleration:
ANGULAR DISPLACEMENT
ANGULAR VELOCITY
ANGULAR ACCELERATION
TIME
8.3 The Equations of Rotational Kinematics
8.3 The 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 that are given for any of the fivekinematic variables.
4. Verify that the information contains values for at least threeof the five kinematic variables. Select the appropriate equation.
5. When the motion is divided into segments, remember thatthe final angularvelocity 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.
8.3 The Equations of Rotational Kinematics
Example 5 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.
8.3 The Equations of Rotational Kinematics
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
8.4 Angular Variables and Tangential Variables
t
rt
r
t
svT
t
rad/s)in ( rvT
8.4 Angular Variables and Tangential Variables
t
rt
rr
t
vva ooToTT
to
)rad/sin ( 2raT
8.4 Angular Variables and Tangential Variables
Example 6 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.
8.4 Angular Variables and Tangential Variables
srad 8.40rev 1
rad 2
s
rev 50.6
sm122srad8.40m 3.00 rvT
8.4 Angular Variables and Tangential Variables
22 sm5.24srad17.8m 3.00 raT
22
srad 17.8rev 1
rad 2
s
rev 30.1
8.5 Centripetal Acceleration and Tangential Acceleration
rad/s)in ( 2
22
r
r
r
r
va Tc
8.5 Centripetal Acceleration and Tangential Acceleration
Example 7 A Discus Thrower
Starting from rest, the throweraccelerates the discus to a finalangular speed of +15.0 rad/s ina time of 0.270 s before releasing it.During the acceleration, the discusmoves in a circular arc of radius0.810 m.
Find the magnitude of the totalacceleration.
8.5 Centripetal Acceleration and Tangential 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
8.6 Rolling Motion
rv
The tangential speed of apoint on the outer edge ofthe tire is equal to the speedof the car over the ground.
ra
8.6 Rolling Motion
Example 8 An Accelerating Car
Starting from rest, the car acceleratesfor 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
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.