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Causes of Rotation
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.
F  m  a
  I 

Sum the torques

Torque replaces force

Inertia replaces mass

Angular acceleration replaces linear acceleration
Inertia

Also known as: Rotational Inertia or Moment of
Inertia

The mass property of a rigid body that determines
the torque needed for a desired angular
acceleration about an axis of rotation.

Tendency to resist angular acceleration

Depends on the shape of the body and may be
different around different axes of rotation.

Objects will have a rotational inertia based on its
size, mass, and axis of rotation

Similar to mass

Unit: kg m²
Point Mass Inertia

Think ball on a string

Rotational Inertia or Moment of Inertia

Similar to mass

r – radial distance from axis of rotation

Unit: kg m²

Systems can consist of many masses
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.
Rigid Object Moment of Inertia
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 of 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
See it!
2
1 2
 mr
2
Parallel Axis Theorem – Rigid
Object

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.

r – represents center of mass position to pivot
I  I cm  mr
2
Parallel Axis Practice

A 300 g rod (L = 75 cm) pivots about a
point, 20 cm from its left end. The
rotational inertia about its center of
mass is given by the expression
1/12 ML². What is the rotational inertia
of the rod about its pivot?
0.023 kg·m²
Pulley System

A 1.5 kg mass is attached to a pulley
(ignore friction) with inertia of
0.002 kgm² and radius .03 m. The mass
is released from rest, what is the
acceleration of the falling mass?
Atwood System

Released from rest, determine the tension in the
string and the acceleration of the vertical Atwood
system.

Ignore effects of the pulley

𝑚2 = 800 g

𝑚1 = 400 g
Atwood System w/Inertia

Released from rest, determine the tension in each
string and the acceleration of the vertical Atwood
system.

The pulley:

I = 0.0001 kg·m²

R = 3 cm

𝑚2 = 800 g

𝑚1 = 400 g
Ignoring Pulley
3.27 m/s/s
5.23 N
Rotational Inertia
Point Masses
Rigid objects

Rotational inertia of
a set of masses
about an axis of
rotation

Integration limits
from axis of rotation
to end points of
rotating object

No negative r values

May have negative r
integration limits
I   mr
2
I   r dm
2
dm   dr
  mass / length