Causes of Rotation
AP Physics
SUM THE TORQUES
Newton’s Second Law (with
rotation)

How much torque is needed to rotate?

You can sum the forces acting on an object and
apply Newton’s Second Law for linear motion.

Newton’s Second Law can be applied to rotational
motion as well, using rotational quantities.
  I  

Sum the torques

Torque replaces force

Inertia replaces mass

Angular acceleration replaces linear acceleration
Inertia

Rotational Inertia or Moment of
Inertia

Similar to mass

r – radial distance from axis of
rotation

Unit: kg m²
I   mr
2
Two weights on a bar

Two ‘weights’ of mass 5.0 kg and 7.0 kg
are mounted 4.0 m apart on a light rod
(ignore mass). Calculate the moment of
inertia of the system when rotated about
an axis halfway between the weights.
48 kg m²
What about angular acceleration?
Continue…

Using the same system, calculate the
moment of inertia of the system when
rotated about an axis 0.5 m to the left
of the 5.0 kg ‘weight’.
143 kg m²
Inertia and Rolling

Inertia and angular acceleration are inversely related (mass and
acceleration are inversely related).
  I 

Objects rolling with low inertia will have high angular acceleration
(low mass requires less force to cause acceleration).


Easier to change speed.
Objects rolling with high inertia will have low angular acceleration
(high mass requires more force to cause acceleration).

Harder to change speed.
Rolling Ring and Disk


A disk and ring of equal mass
and radius rolling down a
ramp from rest:

Disk gets to bottom faster.

Disk has greater
acceleration.
A disk and ring or equal mass
and radius rolling up a ramp
with same initial speed:

Ring will roll higher.

Ring has lower acceleration,
and slows at a lower rate.
I ring  mr
I disk
2
1 2
 mr
2
Various Moments of Inertia
Parallel Axis Theorem

The moments of inertia of rigid objects with simple geometry
are relatively easy to calculate provided the rotation axis
coincides with an axis of symmetry.

The calculation about an arbitrary axis can be found by
adding the rotational inertia about the center of mass and
the rotational inertia of the center of mass about the arbitrary
axis.

The arbitrary axis must be parallel to the center of mass axis of
rotation.
I  I cm  mr
2