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Rotation of Rigid Bodies
Ms. Mikaela Irene Fudolig
Physics 71 Lecture
Rigid Bodies and Rotation
Rigid bodiesbodies with a definite size and shapedo not deform
Rotationfor now, we consider rotation about a fixed axis
Example 1
Identify the axis of rotation for each of the following rotating objects:
Describing Rotational Motion
Use polar coordinates, (r,)!
s r is in
RADIANS!
Signs of :
+ if rotation is CCW
– if rotation is CLOCKWISE
Exercise 1
A girl is on a merry-go-round, 5 meters from the center of the circular platform. What is her angular displacement if she travels a total distance of 2 meters after some time t?
Angular velocity,
Analogous to the linear velocity, vRate of change in angular displacement
Angular velocity,
Average angular velocityInstantaneous angular velocityAngular speed
magnitude of the angular velocity
Average Angular Velocity
The average angular velocity over a time interval t = tf - ti is
ˆ av kt
*assumes
that the rotation axis is the z-axis.*
(Instantaneous) Angular velocity
rate of change of the angular displacement
VECTOR!
ˆ d
kdt
*assumes
that the rotation axis is the z-axis.*
Angular velocity,
Direction along the axis:UPWARDS (+), if
rotation is CCWDOWNWARDS (-), if
rotation is CLOCKWISE
Angular velocity,
CAUTION:
“Clockwise” and “CCW” are NOT directions for the ANGULAR VELOCITY!!!
Angular speed,
magnitude of the angular velocity
d
dt
Angular acceleration,
Analogous to the linear acceleration, aHow fast the angular velocity changes
2
2
ˆ
ˆ
dk
dt
dk
dt
*assumes that the rotation axis is the z-axis.*
Angular acceleration,
Direction along the axis:SAME DIRECTION
AS , if rotation SPEEDS UP
OPPOSITE DIRECTION AS , if rotation SLOWS DOWN
Exercise 2
You put a DVD in the DVD drive of your laptop, and it starts rotating clockwise.Draw a diagram of the rotating DVD, indicating
direction of rotation.Draw the direction of the angular velocity
vector.Draw the direction of the angular acceleration
vector.
Linear vs. Angular Quantities
Linear Angular
x,s
v
a
Motion with Constant Angular Acceleration
2 20
0
2
2t
Analogous to motion with constant linear acceleration (aka Chapter 2)
0
20
1
2
t
t t
Exercise 3
Y&F ex.9.3. A DVD is slowing to a stop. At t=0, its angular velocity is 25.7rad/s and its angular acceleration is a constant –10.0rad/s2.What is the disc’s angular velocity at
t=0.300s?What angle does the line PQ make with the
+x-axis at t=0.300s (I.e., what is (t=0.300s)?
Linear Kinematics of a rotating particle
A particle in a rotating body moves in a circle:
Linear/tangential velocity of a rotating particle
From s=r, we can get the expression for the linear speed of a rotating particle a constant distance r from the axis of rotation:
v rThe direction of the linear velocity vector is TANGENT TO ITS CIRCULAR PATH, parallel to the infinitesimal displacements!
Acceleration of a rotating particle
Tangentialchange in linear speed
Radialchange in direction of the linear velocity
Tangential acceleration
The tangential acceleration of a rotating particle a distance r from the axis of rotation is:
tan
da r r
dt
Direction: parallel or antiparallel to the linear velocity
Radial acceleration
The radial acceleration of a rotating particle a distance r from the axis of rotation is:
22
rad
va r
r
Direction: pointing towards the axis of rotation
Exercise 4Y&F 9.26. An electric ceiling fan with
blades 0.750m in diameter is rotating with constant angular acceleration 0.900rev/s2 about a fixed axis with an initial angular velocity of 0.250rev/s. After 2.00s:what is the angular velocity of the fan?what is the linear speed of the tip of a fan
blade?
Exercise 4
Y&F 9.26. A electric ceiling fan with blades 0.750m in diameter is rotating with constant angular acceleration 0.900rev/s2 about a fixed axis with an initial angular velocity of 0.250rev/s. After 2.00s:what is the radial acceleration?what is the tangential acceleration?