You are viewing an old version of this page. View the current version.

Compare with Current View Page History

« Previous Version 63 Next »

Unknown macro: {table}
Unknown macro: {tr}
Unknown macro: {td}
Error formatting macro: live-template: java.lang.NullPointerException
Unknown macro: {td}

Moment of Inertia

A measure of the effort required to change that object's rotational velocity about a specified axis of rotation.

MotivationforConcept"> Motivation for Concept

MathematicalDefinition"> Mathematical Definition

CalculatingMomentofInertia"> Calculating Moment of Inertia

TheParallelAxisTheorem"> The Parallel Axis Theorem

StatementoftheTheorem"> Statement of the Theorem

The parallel axis theorem states that if the moment of inertia of a rigid body about an axis passing through the body's center of mass is Icm then the moment of inertia of the body about any parallel axis can be found by evaluating the sum:

Unknown macro: {latex}

\begin

Unknown macro: {large}

[ I_

Unknown macro: {||}

= I_

Unknown macro: {cm}

+ Md^

Unknown macro: {2}

] \end

where d is the (shortest) distance between the original center of mass axis and the new parallel axis.

ComplexObjectsasaSumofSimpleConstituents"> Complex Objects as a Sum of Simple Constituents

The principle utility of the parallel axis theorem is in quickly finding the moment of inertia of complicated objects. For example, suppose we were asked to find the moment of inertia of an object created by screwing two hollow spheres of radius R and mass Ms to the end of a thin rod of length L and mass Mr. If the object is rotated about the center of the rod, then the total moment of inertia is found by adding the contributions from the rod to that from the spheres. From the table above, we can see that the rod contributes:

Unknown macro: {latex}

\begin

Unknown macro: {large}

[ I_

Unknown macro: {r}

= \frac

Unknown macro: {1}
Unknown macro: {12}

M_

L^

Unknown macro: {2}

]\end

Since the centers of the spheres are a distance L/2+R away from the axis of rotation of the composite object, they each contribute:

Unknown macro: {latex}

\begin

Unknown macro: {large}

[ I_

Unknown macro: {s}

= \frac

Unknown macro: {2}
Unknown macro: {3}

M_

R^

Unknown macro: {2}

+ M_

Unknown macro: {s}

\left(\frac

Unknown macro: {L}

+R\right)^

Unknown macro: {2}

] \end

so the total moment of inertia is:

Unknown macro: {latex}

\begin

Unknown macro: {large}

[ I_

Unknown macro: {rm obj}

= I_

Unknown macro: {r}

+2I_

Unknown macro: {s}

= \frac

Unknown macro: {1}
Unknown macro: {12}

M_

L^

Unknown macro: {2}

+ \frac

Unknown macro: {4}
Unknown macro: {3}

M_

Unknown macro: {s}

R^

+ 2M_

Unknown macro: {s}

\left(\frac

Unknown macro: {L}
Unknown macro: {2}

+R\right)^

]\end


Error formatting macro: live-template: java.lang.NullPointerException
  • No labels