Lecture 5-3C: Moment of Inertia & Shape

Torque and angular acceleration

moment of inertia (units kg m2)

moment of inertia = distribution of mass in

space

distance perpendicular to

rotation axis

masses in system

I

distance perpendicular to

rotation axis

integral over mass

I

Moments of mass

Total mass

Center of mass

Moment of Inertia (mass distribution)

Moments of inertia for various shapes

ring or hollow cylinder

disk or solid cylinder

solid sphere

stick or rod R

R R

L

plate

A B

Rotation axis is important

Offset axes

Offset axes

Parallel Axis Theorem: For any axis offset from the center of mass (COM): d

perpendicular separation of a

parallel rotation axis

measured from axis passing through center of mass

Building more complicated objects

Moments of inertia for complex shapes can be built up by adding up component parts

1 3 2

2 1

3

model of acetone molecule by Ben Mills: https://commons.wikimedia.org/wiki/File:Acetone-3D-balls.png image of tennis racket by Utcursch: https://en.wikipedia.org/wiki/Glossary_of_tennis_terms#/media/File:Babolat_pure_drive_plus.jpg video of 433 Eros by NEAR Project (JHU/APL): http://nssdc.gsfc.nasa.gov/planetary/mission/near/near_eros_anim.html

Summary The moment of inertia is the mechanical resistance

to torque and measures the spatial distribution of mass of an object:

Moment of inertia only depends on distribution of mass perpendicular to rotation axis, and on the orientation and location of rotation axis

Rotation about an axis offset from center of mass can be computed from parallel axis theorem