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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
8.2 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
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)
8.3 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
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
8.5 Centripetal Acceleration and Tangential Acceleration
rad/s)in ( 2
22
r
r
r
r
va Tc
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.
Problems to be solved
• 8.31, 8.34, 8.49, 8.62, 8.70, 8.73
• B8.1 Convert to degrees (a) 0.1rad (b)π/4rad.Ans: (a) 5.73º (b) 45º
• B8.2 Convert to radians (a) 53⁰ (b) 120⁰.Ans: (a) 0.93rad (b) 2.09rad
• B8.3 The speeds of centrifuges are limited in part by the strength of the materials used in their construction. A centrifuge spins a 10g sample at a radius of 0.05m at 60 000 revolutions per minute. (a) What force must the centrifuge exert on the sample? (b) What is the mass of a sample at rest with a weight equal to this force? Ans: (a) 19739.21N (b) 2014.20kg
• B8.4 A centrifuge used to separate blood cells from blood plasma rotates at 55 rotations per second. What is the acceleration at the centre of a centrifuge tube 8.0cm from the axis of rotation?
Ans: 9.55×103m/s2.
