   Chapter 14, Problem 42RE

Chapter
Section
Textbook Problem

Finding Moments of Inertia and Radii of GyrationIn Exercises 41 and 42, find I x , I y , I 0 , x ¯ ¯ , and y ¯ ¯ for the lamina bounded by the graphs of the equations. y = 4 − x 2 ,     y = 0 ,     x > 0 ,     ρ = k y

To determine

To calculate: The value of Ix,Iy,I0,x¯¯,y¯¯ for the lamina bounded by the graphs of the equation y=4x2,y=0,x>0.

Explanation

Given:

Equation of the graph:

y=4x2,y=0,x>0

The lamina has density ρ=ky.

Formula used:

Along x-axis, the Moment of inertia is: Ix=0204x2y2ρdydx

Along y-axis, the Moment of inertia is: Iy=0204x2x2ρdydx

By perpendicular axis theorem, I0=Ix+Iy

Mass M=0204x2ρdydx

Calculation:

Moment of inertia along x-axis is:

Ix=0204x2y2ρdydx=0204x2ky3dydx

(After first integration)

Ix=1402ky4|04x2dx

Ix=1402k(4x2)4dx(After second integration)=16,384315k

Moment of inertia along y-axis is:

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