Moments of Inertia

Moments of inertia for radially symmetric objects

Consider objects of uniform density p and with the following shapes:

(a) L = 1 m
r = 0.02 m
(b) L = 0.0016 m
r = 0.5 m
(c) L = 0.0016 m
r1 = 0.5 m
r2 = 0.6 m

For the object above, let A be the area of a cross-section taken in the plane perpendicular to the x-axis, for example, A = pi*r^2 for the first example. Then the mass of the object is

Question 1

Compute the mass of each of the three objects shown above.

Question 2

The contribution to the moment of inertia from a thin shell of radius r is
r^2 (2*pi*r*p) dr
.
(Argue geometrically why this should be true.) Thinking of the previous object as being composed of thin shells,
  1. find the moment of inertia for each of the objects by integrating

    over the appropriate range of r values.
  2. Which object is hardest to rotate? Easiest? Compare the mass of the objects with their rotational inertia. How easy is it to move the most massive object?

Index

More: Planar Symmetric Objects

Next: Cantilevered Beams

Up: Introduction

Previous: Centers of Mass and Centroids

Jennifer Powell<jpowell@geom.umn.edu>
Fati Liamidi<liamidi@geom.umn.edu>