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Rolling Inertia
 Solid Cylinder or Hollow Cylinder?
 Greatest mass?
 Greatest acceleration?
 Greatest rotational inertia?
 Greatest momentum?
 Greatest initial total mechanical energy?
 Greatest kinetic energy at the bottom?
Energy of Rotational Motion
AP PHYSICS
APPLY THE CONCEPTS OF ENERGY
CONSERVATION TO ROTATIONAL MOTION IN
ORDER TO DETERMINE THE SPEED OF A
ROTATING OBJECT
Energy
Potential
Rotational Kinetic
 Same energy
 All rotating objects have
 h is measured with respect
to center of mass
 Unit: J
U g  mgh
Total Mechanical Energy is still
conserved.
rotational energy
 If an object spins in place, it
will have only rotational K
 While an object rolls along a
surface, it will have
rotational K and
translational K
2
 Unit: J
R
1
K  I
2
Pulley System
Ignoring Pulley
Pulley with Rotational K
 Acceleration of the system
 Acceleration of the system
determined by summing
forces
 Ug and translational K
present in the system
 How fast will m2 be
traveling after falling a
distance h?

Linear quantities only
determined by summing
torques and forces
 Ug, translational K, and
rotational K present in the
system
 How fast will m2 be
traveling after falling a
distance h?

Linear and angular quantities
Simple Pendulum
 A pendulum swings from a pivot point. If the
pendulum is released from rest, determine the speed
of the pendulum bob through equilibrium.
 You can solve using rotational energy or
translational energy (they will be the same).
 Pendulum bob is considered a point mass.
Ball rolling down ramp
 Rigid object
 What kind of energy is present in the system?
 Complete the intro. problem (hoop) again, using
energy conservation methods only.
Atwood System
 The masses in the system are released from rest. If
the masses are 800 g and 400 g respectively, and
the 3.0 cm radius pulley has a rotational inertia of
0.0001 kg·m², using energy conservation,
determine the velocity of the masses after traveling
a vertical distance of 30 cm.
Momentum of Rotational
Motion
AP PHYSICS
MOMENTUM
What is Conserved?
Momentum
 Angular momentum: when mass is rotating
 Angular speed must be in terms of radians/sec
 Cross product: quantities must be perpendicular
(sin)
 Unit: kg m²/s
  
Lrp
Point mass (ex: mass on string)

L  I 
Rotating mass (ex: ball rolling)
Angular momentum is conserved.
• When net torque is ZERO, L is constant.
• Separate from linear momentum
Conservation
 Angular momentum is conserved (separate
from linear momentum)
 Net
torque must be zero if angular momentum is
to be conserved
∑𝜏 = 0
THEN
∆𝐿 = 0
𝐿1 = 𝐿2
Angular Momentum Practice 1
 A mass is attached to the end of a string that revolves in a
circle on a frictionless tabletop. The other end of the string
passes through a hole in the table. The mass initially
revolves at a speed of 2.4 m/s in a circle of radius = 0.8 m.
The string is slowly pulled through the hole to a radius =
0.48 m. What is the new speed of the mass?
Angular Momentum Practice 2
 A rod (mass = 3m and length = l) hangs from a
pivot at its end. A ball (m) strikes the rod
(I = 1/3 Ml²), traveling at a velocity (v), and sticks
to the opposite end. Determine the speed at which
the system begins to rise.
Newton’s Second Law
 Many forms of the same equation.
 Think of the angular equivalent to the linear
equation
 F  ma
p
 F  t
  I 
L
   t
Work
Linear
Rotational
 Force x distance (dot
 Torque x angular
product - parallel)
 Unit: J W  F  r
displacement (dot
product) W    
 Unit: J
W
F
r
W


Power
Linear
Rotational
 Force x velocity (dot
 Torque x angular speed
product - parallel)
 Unit: W
(dot product)
 Unit: W
P  F v
P   
Newton’s Second Law w/Calc
 Many forms of the same equation.
 Think of the angular equivalent to the linear
equation
 F  ma
  I
dp
 F  dt
dL
  dt
Work w/Calc
Linear
Rotational
 Force x distance
 Torque x angular
 Unit: J
displacement
 Unit: J

x
W   F  dx
xo
dW
F
dr
W     d
o
dW

d
Power w/Calc
Linear
Rotational
 Force x velocity
 Torque x angular speed
 Unit: W
 Unit: W
dW
P
dt
t
W   P  dt
to
Rotational Motion Review Problem
 The system pictured is released from rest. The pulley
is frictionless and so is the surface of the incline. After
the system moves a distance of 50 cm, determine each
of the following for the pulley:
𝑚1 = 2 𝑘𝑔 𝑚2 = 5 𝑘𝑔 𝜃 = 30°
1.
2.
3.
4.
Angular speed
Angular acceleration
Angular momentum
Rotational kinetic energy
𝑝𝑢𝑙𝑙𝑒𝑦
M = 100 g R = 10 cm I =
1
2
2𝑀𝑅