Phys 1111

1
Phys 1111

• Chapter 9
• Rotational Kinematics and Dynamics

2
Measure of Rotation
r
?s
??
Where ? is in radian (rad)
Converting degree to radian and radian to degree
3
Measure of Rotation

• Example 1
• Convert each of the following angles to radian
  measure 60o, 250o, -45o
• Convert each of the following angles (given in
  radian measure) in to degree 4?/6, 9 ? /5, - ?
  /12
• As a turn table rotates 60o its outer most edge
  travels 0.20m find the radius of the turn table.

4
Angular Velocity and Angular Acceleration
r
?t
?s
??
Average angular speed
As the object moves through the arc length ?s in
the interval ?t
v is the tangential speed of the particle
5
Angular Velocity and Angular Acceleration
?2
r
?t
?s
??
?1
Average angular acceleration
The tangential acceleration
6
Angular Velocity and Angular Acceleration
Period (T) Time for one complete
rotation. Frequency (f) the number of turns per
unit time.
If the particle makes n number of turns in a time
?t the frequency is
7
Angular Velocity and Angular Acceleration
Example 2 The rotor on a helicopter turns at an
angular speed of 320 revolutions per minute (rpm)
express this in radians per second.

• Example 3
• A floppy disk in a computer rotates from rest up
  to an angular speed of 31.4 rad/s in a time of
  0.9s.
• What is the angular acceleration of the disk,
  assuming the angular acceleration is uniform?
• If the radius of the disk is 4.45cm , find the
  final linear speed of the microbe riding on the
  rim of the disk.
• What are the tangential and radial accelerations
  of the microbe at this time?

8
Torque and angular acceleration
If the force on a rotating object has a
tangential component
Ft
?2
r
Fr
?s
??
?1

• Where
• Ftr torque
• I mr2 moment of inertia

9
Torque and angular acceleration
Note that the equation ? Ia is the rotational
counterpart to Newton's second law F ma
Torque on a rotating object
l
Consider a rotating rigid body
10
Torque and angular acceleration

• Example 4
• Calculate the net torque on the beam in the
  figure below about
• an axis through O perpendicular to the page, and
• an axis through C, perpendicular to the page

25N
30o
C
O
20o
45o
2m
10N
30N
4m
11
Torque and angular acceleration
Moment of inertia
Axis of rotation
Moment of inertia depends on the axis of rotation
m1
m2
r1
r2
r3
m3
The moment of Inertia of a body depends on the
axis of rotation and on the manner in which the
mass is distributed
12
Torque and angular acceleration

• Example 5
• Four masses are held in position at the corners
  of a rectangle by light rods as shown in the
  figure below. Find the moment of inertia of the
  system about
• the x axis
• The y axis
• an axis through O and perpendicular to the page.

13
Torque and angular acceleration
Moment of inertia of simple solids with uniform
density
14
Torque and angular acceleration

• Note that
• The moment of inertia of the same solid is in
  general different about different axis
• The moment of Inertia has units of
  (mass)x(distance)2 . Such as kg.m2, g.cm2,
• If objects of equal mass rotate about the same
  axis, the moment of inertia will be greater for
  object whose mass is distributed farther from the
  axis.

15
Torque and angular acceleration
Example 6 When a torque of 32.0N.m is applied to
a certain wheel, the wheel acquires an angular
acceleration of 25.0rad/s2. What is the moment of
inertia of the wheel?
16
Rotational Kinetic energy
For a point object traveling in a circle with a
speed v
r
For a rigid body
17
Rotational Kinetic Energy
?
m1
m2
r1
r2
r3
m3
18
Rolling Object
Rotational Kinetic energy
?
v
For a rolling object without slipping
?
R
vcm
19
Rotational Kinetic energy
Example 7 A ball of mass M and radius R starts
from rest at a height of 2.00m and rolls down a
30.0o slope, as shown in the figure below. What
are the linear and angular speeds of the ball
when it leaves the incline? Assume that the ball
rolls without slipping.
20
Angular Momentum
Consider an object of mass m positioned a
distance of r away from the center of rotation.
Under the action of a constant torque on the
object, its angular speed of will increase from
?1 to ?2 in a time of ?t. Thus, we can write
m
r
Definition of angular momentum (L)
Then we can write
or
21
Angular Momentum
When the net external torque ?? 0 we see that
?L 0, i.e.
or
For a system consisting of many objects
The total angular momentum of the system is
conserved if the total external torque on the
system is zero. That is, when ?? 0, the initial
angular momentum equals the final angular
momentum.
22
Angular Momentum
Example 8 A light rod 1.00m length rotates about
an axis perpendicular to its length and through
its center as shown in the figure below. Two
particles of masses 4.00kg and 3.00kg are
connected to the ends of the rod. What is the
angular momentum of the system if the speed of
each particle is 5.0m/s
4.00kg
3.00kg
23
Angular Momentum
Example 9 A solid, horizontal cylinder of mass
10.0kg and radius 1.00m rotates with an angular
speed of 7.00rad/s about a fixed vertical axis
through its center. A 0.250kg piece of putty is
dropped vertically onto the cylinder at a point
0.900m from the center of rotation, and sticks to
the cylinder. Determine the final angular speed
of the system.
24
Analogies
25
Exercises

• 9.11, 9.19, 9.27, 9.32, 9.38, 9.39