   Chapter 14.4, Problem 12E

Chapter
Section
Textbook Problem

ConjectureUse the result of Exercise 11 to make a conjecture about the change in the center of mass when a lamina of constant density is translated c units horizontally or d units vertically. Is the conjecture true when the density is not constant? Explain.

To determine

To calculate: The conjecture about the change in center of mass when a lamina of a constant density is translated c units horizontally or d units vertically.

Explanation

Given:

The solution for the Exercise 11(a) For Square with vertices (5,0), (a+5,0), (5,a), (a+5,a) and density ρ=k.

The center of the mass is:

(x¯,y¯)=(a2+5,a2)

11(b) For Square with vertices (5,0), (a+5,0), (5,a), (a+5,a) and density ρ=ky.

The center of the mass is:

(x¯,y¯)=(a2+5,2a3).

11(c) For Square with vertices (5,0), (a+5,0), (5,a), (a+5,a) and density ρ=kx.

The center of the mass is:

(x¯,y¯)=(2(a2+15a+75)3(a+10),a2)

Formula used:

Mass of planar lamina is:

m=Rρ(x,y)dA

Moment of mass of variable density planar lamina is:

Mx=R(y)p(x,y)dAMy=R(x)p(x,y)dA

The center of mass.

(x¯,y¯)=(Mym,Mxm)

The integral of, xndx=xn+1n+1+c.

Calculation:

For a square with vertices (0,0), (a,0), (0,a), (a,a) and density ρ=k.

The mass of the lamina is:

To find the center of the mass, find the moment of inertia about both axis

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