Rotational Equilibrium and
Dynamics
Rotational Energy &
Rotational Dynamics
Problems
Rotational Kinetic Energy


You have learned previously that the
mechanical energy of an object includes its
translational kinetic energy and its
potential energy.
This approach did not consider the possibility
that objects could have rotational motion
along with translational motion.
Rotational Kinetic Energy


Rotating objects possess kinetic energy
associated with their angular speed.
We call this energy rotational kinetic
energy.
KErot
1 2
 I
2
(rotational kinetic energy)=1/2(moment of inertia)x(angular speed)2
Mechanical Energy

Taking rotational motion into account, the
mechanical energy of a system is the sum
total of the translational kinetic energy,
gravitational potential energy, and
rotational kinetic energy.
ME  KEtrans  KErot  PEg
1 2 1 2
ME  mv  I   mgh
2
2
Conservation of Mechanical Energy

Recall that the total energy of a system is
conserved.
MEi  ME f
Analysis of the rolling cans:





Racing identical pop cans. What happens?
What energies do the cans have at the top of
the ramp?
At the bottom?
Which has more translational KE?
Which has more rotational KE?
Linear v. Angular Quantities
Linear
Angular
Displacement
x
q
Velocity
v

Acceleration
a
a
Inertia
m
I
Momentum
p=mv
L=I
Kinetic Energy
½ m v2
½ I2
Sample Problem #1

A solid ball with a mass of 4.10 kg and a
radius of 0.050 m starts from rest at a height
of 2.00 m and rolls down a 30.0o slope. What
is the translational speed of the ball when it
leaves the incline?
Sample Problem #2

As Halley’s comet orbits the sun, its distance
from the sun changes dramatically, from
8.8x1010 m to 5.2x1012 m. If the comet’s
speed at closest approach is 5.4x104 m/s,
what is its speed when it is farthest from the
sun if angular momentum is conserved?
Sample Problem #3

Assume that a yo-yo has a mass of
6.00x10-2 kg. If a yo-yo descends from a
height of 0.600 m down a vertical string and
had a linear speed of 1.80 m/s by the time it
reached the bottom of the string. If its final
angular speed was 82.6 rad/s, what was the
yo-yo’s moment of inertia?