Download Rotational Motion

Rotational Motion
Angular Quantities
• Angular Displacement
• Angular Speed


t
   f   i
d

dt
• Angular Acceleration


t
d d 

 2
dt
dt
2
Linear to Angular
• Angle to Distance
s  r
• Angular to Linear Velocity
vt    r
• Angular to Linear Acceleration
at    r

r
s
Kinematics
• For each kinematic equation there is an
angular analog.
v  12 v0  v 
  12  0   
v  v0  at
  0   t
x  x0  vt  at
1
2
2
  0   t   t
1
2
2
v  v  2a x  x0      2    0 
2
2
0
2
2
0
Bike Ride
• A bicycle wheel with radius 0.91 m is
spinning at 2.2 revolutions per second. If
the wheel stops in 50 m, what is the average
angular deceleration?

Torque
• Torque - Force’s effectiveness in altering
rotation motion. (Also called the moment
of the force.)
F
Pivot
r
Torque depends on
Distance from pivot
Magnitude of applied force
Direction of applied force
  rF sin 
  
  r F
Units (N·m)

Diving
• A 85 kg man stands on a diving board. If
he is 2.5 m from the pivot point, what
torque is he exerting on the board?
Moment of Inertia
• From Newton’s 2nd Law
F  ma

 m r
r
  mr 
2
  I
Moment of Inertia
One particle
I  mr
2
Multiple particles
I total   mi ri
2
Weighted Bar
• What is the moment of inertia for the object
illustrated below if it spins about its center
of mass?
0.6 m
0.8 m
8 kg
2 kg
2.4 m
Continuous Media
• For Continuous Media the
moment of inertia is
I   r dm
2
• Ex. What is the moment of
inertia for the washer about its
axis if the inner radius is 1.0
cm and the outer radius is 3.0
cm and the mass is 20 g?
b
a
Atwood’s Machine
• Two masses are attached
to a string and placed
over a pulley of mass 2.0
kg and radius 15 cm.
• What is the acceleration
of the masses if m1 = 3.0
kg and m2 = 2.0 kg?
m2
m1
Parallel-Axis
Theorem
• If the moment of inertia is
know w.r.t. the center of mass,
then by shifting the pivot point
the new moment of inertia is
I  I cm  Mh
2
• From the washer in the
previous problem what is the
moment of inertia about the
edge of the washer?
New Pivot
h
CM
Rotational Energy
• Rotational equivalent to kinetic energy
K rot  I
1
2
K total  I
1
2
K total 
1
2
I
2

2
cm

 MR 
2
Pivot
2
Ktotal  I cm  M R 
1
2
2
K total  I cm  Mv
1
2
2
2
1
2
1
2
2
Ktotal  K rot  Ktrans
Race Down the
Slope
• A block and a ball move down an inclined
plane. The block is able to slide while the
ball has enough friction to roll.
What are the velocities when they reach the
bottom of a 0.25m high
slope assuming frictional
loses are negligible?