Chapter 10 Rotation

1
Chapter 10 - Rotation

• Definitions
• Angular Displacement
• Angular Speed and Velocity
• Angular Acceleration
• Relation to linear quantities
• Rolling Motion
• Constant Angular Acceleration
• Torque
• Vector directions
• Moment Arm
• Newtons 2nd Law for Rotation
• Calculating Rotational Inertia
• Moment of inertia
• Using the table
• Parallel Axis Theorem
• Perpendicular Axis Theorem
• Conservation of Angular Momentum
• Rotational Kinetic Energy
• Work and Rotational Kinetic Energy

2
Radius vs. position vector
3
Kinematics Memory Aid
4
Velocity

• Average velocity
• Instantaneous velocity

5
Angular Displacement
6
Angular Velocity

• Average angular velocity
• Instantaneous angular velocity

7
Acceleration

• Average acceleration
• Instantaneous acceleration

8
Angular Acceleration

• Average angular acceleration
• Instantaneous angular acceleration

9
Rotational Kinematics Memory Aid
10
Converting angular to linear quantities
velocity
tangential acceleration
11
Radial acceleration
12
Frequency vs. angular velocity

• Frequency
• Cycles per time interval
• Revolutions per time interval
• Hertz
• Angular velocity
• Radians per time interval
• Sometimes called angular frequency
• Radians/sec

13
Constant Acceleration
14
Constant Angular Acceleration
15
Problem 1

• A record player is spinning at 33.3 rpm. How far
  does it turn in 2 seconds.
• The motor is shut off. The record player spins
  down in 20 seconds (assume constant
  deceleration).
• What is the angular acceleration?
• How far does it turn during this coast down?

16
Vector nature of angular quantities
17
Rolling without slipping
18
Problem 2

• A cylinder of radius 12 cm starts from rest and
  rotates about its axis with a constant angular
  acceleration of 5.0 rad/s2. At t 3.0 sec, what
  is
• Its angular velocity
• The linear speed of the point on the rim
• The radial and tangential components of
  acceleration of a point on the rim.

19
Torque causes angular acceleration

• Torque is the moment of the force about an axis
• Product of a force and a lever arm
• Rotational Analog to Newtons 2nd Law

20
What if the force is not perpendicular?
21
Vector Multiplication Cross Product
22
Right Hand Rule II
23
Vector Multiplication Scalar Product
24
The Torque Vector
R
R
25
Problem 3

• Find the net torque on the wheel about the center
  axle

26
Rotational Inertia
Moment of inertia for a single particle
27
General Moment of Inertia
28
Problem 4

• Three equal point masses are rotating about the
  origin at 2 rad/sec.
• The masses are located at (4m, 0) (0, 4m) and
  (4m, 4m).
• Each mass is 2 kg
• Find the moment of inertia.

29
Moment of inertia of a uniform cylinder
30
Moments of Inertia of various objects
See Figure in book
If particular axis is not in the table, use the
parallel axis theorem
31
Problem 5

• A disk with radius, R, and mass, M, is free to
  rotate about its axis. A string is wrapped
  around its circumference with a block of mass, m,
  attached. This block is released from rest and
  falls.
• Find the tension in the string
• Find the acceleration
• Find the velocity after the mass has fallen a
  distance, h.

R
M
m
32
Angular Momentum
If there are no torques
33
Two conservation of angular momentum
demonstrations
34
Precession
35
Keplers 2nd Law

• The Law of Areas
• A line that connects a planet to the sun sweeps
  out equal areas in equal times.

36
Rotational Kinetic Energy
37
Problem 6 - Energy

• A disk with radius, R, and mass, M, is free to
  rotate about its axis. A string is wrapped
  around its circumference with a block of mass, m,
  attached. This block is released from rest and
  falls.
• Find the tension in the string
• Find the acceleration
• Find the velocity after the mass has fallen a
  distance, h.

R
M
m
38
Sphere rolling down a hill
Mass M, initially at rest
Find the velocity at the bottom of the hill?
39
Which is fastest?