   Chapter 14, Problem 41RE

Chapter
Section
Textbook Problem

Finding Moments of Inertia and Radii of Gyration In 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 = 0 ,     y = 2 ,     x = 0 ,     x = 3 ,     ρ = k x

To determine

To calculate: The value of Ix,Iy,I0,x¯¯,y¯¯ for the lamina bounded by the graphs of the equation y=0,y=2,x=0,x=3. The density of the lamina is given as ,ρ=kx

Explanation

Given:

Equation of the graph,

y=0,y=2,x=0,x=3

The lamina has density ,ρ=kx

Formula used:

Moment of inertia along x-axis: Ix=0302y2ρdydx

Moment of inertia along y-axis: Iy=0302x2ρdydx

By perpendicular axis theorem, I0=Ix+Iy

Mass M=0302ρdydx

Calculation:

Take the lamina bounded by graph y=0,y=2,x=0,x=3 into consideration.

Moment of inertia along x-axis is calculated as follows:

Ix=0302y2ρdydx=0302kxy2dydx=0302kxy2dydx=8303kxdx

Integrating further, we get,

Ix=8302kxdx=726k=12k

Moment of inertia along y-axis is calculated as follows:

Iy=0302x2ρdydx=0302kx3dydx=0302k

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