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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