1. Causes of Rotation
Sum the Torques

2. 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.
Sum the torques
Torque replaces force
Inertia replaces mass
Angular acceleration replaces linear acceleration

3. 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

4. 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

5. 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?

6. 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

7. Inertia and Rolling
Inertia and angular acceleration are inversely related (mass and acceleration are inversely related).
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.

8. Rigid Object Moment of Inertia

9. 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.
See it!

10. 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

11. 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 /12 ML². What is the rotational inertia of the rod about its pivot?
0.023 kg·m²

13. Pulley System
A 1.5 kg mass is attached to a pulley (ignore friction) with inertia of kgm² and radius .03 m. The mass is released from rest, what is the acceleration of the falling mass?
3.95 m/s/s 132 rad/s/s

14. Atwood System
Released from rest, determine the tension in the string and the acceleration of the vertical Atwood system.
Ignore effects of the pulley
𝑚 2 =800 g
𝑚 1 =400 g
3.27 m/s/s 5.23 N

15. Atwood System w/Inertia
Released from rest, determine the tension in each string and the acceleration of the vertical Atwood system.
The pulley:
I = kg·m²
R = 3 cm
𝑚 2 =800 g
𝑚 1 =400 g
3.0 m/s/s 5.12 N and 5.44 N
Ignoring Pulley
3.27 m/s/s 5.23 N

17. Rotational Inertia Point Masses Rigid objects
Integration limits from axis of rotation to end points of rotating object
May have negative r integration limits
Rotational inertia of a set of masses about an axis of rotation
No negative r values