   Chapter 16.4, Problem 26E

Chapter
Section
Textbook Problem

Use Exercise 25 to find the moment of inertia of a circular disk of radius a with constant density ρ about a diameter. (Compare with Example 15.4.4.)

To determine

To find: The moment of inertia of a circular disk.

Explanation

Given data:

A circular disk of radius a and constant density ρ .

Formula used:

Write the expression for moment of inertia about y-axis (Iy) .

Iy=ρ3Cx3dy (1)

Here,

ρ is density.

The curve of circular disk is symmetrical curve. Hence, the moment of inertia is symmetric about any two diameters are same. Consider the center of circular disk is located at origin. Therefore, the moment of inertia about diameter is as same moment of inertia about y-axis.

Consider a circular disk C, 0t2π with parametric equations,

x=acost (2)

y=asint (3)

Differentiate equation (2) with respect to t.

ddt(x)=ddt(acost)dxdt=a(sint) {ddt(cost)=sint}dx=asintdt

Differentiate equation (3) with respect to t.

ddt(y)=ddt(asint)dydt=acost {ddt(sint)=cost}dy=acostdt

Substitute acost for x and acostdt for dy in equation (1),

Iy=ρ3C(acost)3(acostdt)=ρ302πa4cos4tdt=13a4ρ02π(cos2t)2dt=13a4ρ02π(12(1+cos2t))2dt{cos2t=12(1+cos2t)}

Expand the equation

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