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Examples of rotational inertia

In this section, I will attempt to describe some simple geometric shapes and provide the formula for the rotational inertia for each shape. This informationis largely based on information gleaned from (External Link) .

Terminology

Unless I specify otherwise, the axis of rotation will be the axis of symmetry, such as at the center of a wheel. Also, unless I specify otherwise,

  • r represents the radius
  • m represents the mass

Facts worth remembering -- Examples of rotational inertia

Thin hollow cylindrical shape or hoop

Think of a can of beans without the beans and without the end caps.

I = m*r^2

Solid cylinder or disk

I = (1/2)*m*r^2

Thick-walled cylindrical tube with open ends, of inner radius r1, and outer radius r2

I = (1/2)*m*(r1^2 + r2^2)

Thin rectangular plate of height h and width w with axis of rotation in the center, perpendicular to the plate

I = (1/12)*m*(h^2 + w^2)

Solid sphere

I = (2/5)*m*r^2

Then hollow spherical shell

I = (2/3)*m*r^2

Thin rod of length L

Axis of rotation is perpendicular to the end of the rod.

I = (1/3)*m*L^2

Thin rectangular plate of width L and height H

Axis of rotation is along the edge of the plate parallel to the H dimension and perpendicular to the width L.

I = (1/3)*m*L^2

Thin rod of length L

Axis of rotation is through the center of the rod.

I = (1/12)*m*L^2

Thin rectangular plate of width L and height H

Axis of rotation is along the center of the plate parallel to the H dimension perpendicular to the width L.

I = (1/12)*m*L^2

Example scenarios

I will apply some of what we have learned to several different scenarios in this section.

The 2x4 scenario

Returning to the earlier example, pick up an eight-foot piece of 2x4 lumber, grasp it near one end, and try swinging it like a baseball bat.

Then grasp it in the center and rotate it as far as you can without hitting your body.

How does the rotational inertia with the axis at the end compare with the rotational inertia with the axis at the center? What is the ratio of the two?

Solution:

Although this may not be a good approximation, we will use the formulas for a thin rectangular plate of height h and width z from (External Link) . (A 2x4 isn't very thin so this may not be a good approximation.)

When rotated around the end,

Iend = (1/3)*(m*h^2) + (1/12)*(m*z^2)

When rotated around the center,

Icen = (1/12)*(m*h^2 + m*z^2)

where

  • Iend is the rotational inertia with the axis of rotation at the end
  • Icen is the rotational inertia with the axis of rotation at the center
  • m is the mass
  • h is the height
  • z is the width (I avoided the use of w because I have been using that character for angular velocity.)

Define the numeric values

Let h = 96 inches and z = 3.75 inches (a 2x4 really isn't 2 inches thick and 4 inches wide)

Let mass = 1kg. We don't know what the mass of a 2.4 is. However, it will cancel out when we compute the ratio of the two cases. We can use any value solong as we don't ascribe any credibility to the absolute rotational inertia value.

Substitute numeric values for symbols

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Source:  OpenStax, Accessible physics concepts for blind students. OpenStax CNX. Oct 02, 2015 Download for free at https://legacy.cnx.org/content/col11294/1.36
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