Chapter 10 - Rotation
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Definitions:
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Angular Displacement
Angular Speed and Velocity
Angular Acceleration
Relation to linear quantities
Rolling Motion
Constant Angular Acceleration
Torque
– Vector directions
– Moment Arm
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Newton’s 2nd Law for Rotation
Calculating Rotational Inertia
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Moment of inertia
Using the table
Parallel Axis Theorem
Perpendicular Axis Theorem
Conservation of Angular Momentum
Rotational Kinetic Energy
Work and Rotational Kinetic Energy

R
Radius vs. position vector
Kinematics Memory Aid
d2x
a 2
dt
dx
dt
dv
dt
x, x
vdt

v, v
a
adt

Forces cause acceleration!!!
Velocity
• Average
velocity
x x 2  x1
v

t
t 2  t1
• Instantaneous
velocity
x dx
v  lim

t  0  t
dt
L
 T 
Angular Displacement
  2  1
Angular Velocity
• Average
angular
velocity
 2  1


t t 2  t1
• Instantaneous
angular
velocity
 d
  lim

t  0 t
dt
 radians   1 
 T    T 
Acceleration
• Average
acceleration
v v 2  v1
a

t
t 2  t1
• Instantaneous
acceleration
v dv
a  lim

t 0 t
dt
L
 T 2 
dv d  dx  d 2 x
a
   2
dt dt  dt  dt
Angular Acceleration
• Average angular
acceleration
 2  1


t
t 2  t1
• Instantaneous
angular
acceleration
 d
  lim

t  0 t
dt
 radians   1 
 T 2    T 2 
d d  d  d 2

   2
dt dt  dt  dt
Rotational Kinematics Memory Aid
, 
d 2
 2
dt
d
dt
d
dt
, 

What causes angular acceleration?
 dt
 dt
Converting angular to linear quantities
 R
velocity
d
d
v
R
 R
dt
dt
tangential acceleration
d2
d 2
a  2  R 2  R
dt
dt
Radial acceleration
v2
aR 
r
aR
R 


R
2
 R2
a  a tan  a R
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
 2 rad 
f

 1 rev 

f
2
1
T
f
Constant Acceleration
v  v0  at
1 2
x  x 0  v 0 t  at
2
v  v  2a  x  x 0 
2
2
0
v  v0
v
2
Constant Angular Acceleration
  rad / s 2 
  4.2 rad / s 2
  0  t
1 2
  0  0 t  t
2
    2    0 
2
2
0
  0

2
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?
Vector nature of angular quantities
Rolling without slipping
v  R
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.
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
Newton’s 2nd Law

  RF
What if the force is not perpendicular?
  R F
  RF
  RFsin 
Vector Multiplication – Cross Product
A  B  A B sin 
ˆi  ˆi  ˆj  ˆj  kˆ  kˆ  0
ˆi  ˆj  kˆ
A  A x ˆi  A y ˆj  A z kˆ
B  Bx ˆi  By ˆj  Bz kˆ
ˆj  kˆ  ˆi
kˆ  ˆi  ˆj
ˆi
A  B  Ax
ˆj
Ay
kˆ
Az
Bx
By
Bz
Right Hand Rule II
ˆi
C  A  B  Ax
Bx
ˆj
Ay
By
kˆ
Az
Bz
Vector Multiplication – Scalar Product
A B  A B cos 
ˆi ˆi  ˆj ˆj  kˆ kˆ  1
ˆi ˆj  ˆi kˆ  ˆj kˆ  0
A  A x ˆi  A y ˆj  A z kˆ
B  Bx ˆi  By ˆj  Bz kˆ
A B  A x Bx  A y By  A z Bz
The Torque Vector
  RF
R
R
  R F sin 
Problem 3
• Find the net torque on
the wheel about the
center axle
Rotational Inertia
  R F sin   R  ma 
  R  mR    mR 2  
  I
Moment of inertia for a single particle
I  mR
2
General Moment of Inertia
n
I   m i R i2
i 1
I   R 2 dm
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.
Moment of inertia of a uniform cylinder
I   R 2 dm
I
dm  dV  RdRddz
R0 2 z

0
4
0
R
1
2
R

RdRd

dz


2

z

MR
0
0 0
4
2
2
Moments of Inertia of various
objects
See Figure in book
If particular axis is not in the table,
use the parallel axis theorem:
I P  I  MR 2
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
Angular Momentum
p  mv
dp
F  ma 
dt
L  I
dL
  I 
dt
If there are no torques:
L  I  I00  constant
Two conservation of angular
momentum demonstrations
Precession
Kepler’s 2nd Law
• The Law of Areas
– A line that connects a
planet to the sun
sweeps out equal areas
in equal times.
1
dA  rvdt
2
v
L  I  mr 2  mrv  Constant
r
1L
dA 
dt
2m
Rotational Kinetic Energy
1
K  mv 2
2
1 2
K  I
2
W  F d
W   d
dW F d
P

F v
dt
dt
dW d
P

 
dt
dt
1 2 1 2
W  If  Ii
2
2
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
Sphere rolling down a hill
Mass = M, initially at rest
Find the velocity
at the bottom of
the hill?
Which is fastest?