Rotational Motion Notes – Physics C
Rotational Kinematics
Kinematics goes Greek: the definitions are the same as for linear kinematics, but the letters are
Greek. Spend about a day discussing and drawing the parallels with what they already know about
kinematics.
= angular displacement
d

and  
(angular velocity)
ave 
dt
t
d

(angular acceleration)
dt
For constant angular acceleration:
  o  t
1
  ot  t 2
2
 2  o2  2
Relationship between linear and angular variables:
s  R
v  R  where R is the radius from the axis of rotation to the point in question.
a  R
Moments of Inertia
Calculating moments of inertia for systems of discrete objects:
I 
m r
i
i i
2
where mi is the mass of each piece and ri is the radius from the axis of rotation
Calculating moments of inertia for continuous objects:

I  r 2dm
Should be able to do this for
- long thin rod about any axis
- hoop
- solid disk about the center
Parallel axis theorem: using the moment of inertia of the object about the center of mass to find
the moment of inertia about an axis that is parallel to the other axis:
I  Icm  md 2 where d is the distance from the center of mass to the axis of rotation.
Newton’s Second Law for Inertia
  I 
the sum of the torques is equal to the product of the moment of inertia and the angular
acceleration;
  rF sin  where r is the distance from the axis to the point of application of force F and  is the
angle between r and F.
Equilibrium:
  0
F  0
Special case of rotation; any axis can be used and choice of axis is
determined by solution method; in other words, choose the axis to make the torque due to unknowns
equal to zero to make solution easier.
Several problems to solve in class:
- single mass attached to a pulley of moment of inertia, I
- Atwood’s machine with two masses and a massive pulley
- Disk/hoop/sphere rolling down an incline: find angular acceleration, friction, etc.
These are usually done as sample problems in any good textbook.
Rotational Energy
The rotational energy is the sum of the linear kinetic energies of each m in the rotating object.
1
1
1
1
2
RKE 
mv
mi (ri )2 
mi ri 2 2  I 2
i i 
2
2
2
2
Solve problems using conservation of energy just as before with RKE as a new term in the
conservation of energy equation




Problems to solve:
- Disk/hoop/sphere rolling down a hill and/or around a loop-the-loop
- Horizontal bar pivoted at one end and released from rest; find angular velocity as it
passes through vertical
- Any rotation problem where the angular acceleration is NOT constant.
Angular Momentum
Use when a collision causes an object to rotate; when the net external force is not zero (due to the
pivot) but the net external torque is zero.
General form: L  r  p Useful forms:
L  I
L  mvr sin 
top: solid object; bottom: moving point mass.
r (length of the radius vector) is the distance from the pivot to the mass m, and  is the angle
between the velocity and the radius vector.
dL
  dt
The time rate of change of the angular momentum of a system about some origin in an
inertial frame equals the net external torque acting on the system about that origin.
Ii i  If f
conservation of angular momentum
Problems to solve:
- ball colliding with a stick (both fixed by a pivot and free)
- ice skater pulling arms in to spin faster (nice demo too, if you have a turn table).