   Chapter 8.3, Problem 35E

Chapter
Section
Textbook Problem

Calculate the moments Mx and My and the center of mass of a lamina with the given density and shape.35. ρ = 6 To determine

To find: The moments Mx, My and the center of mass of a lamina with the given density and shape.

Explanation

Given:

The density is ρ=6

Calculation:

Calculate the area of given shape:

A=14πr2+bh (1)

Substitute 4 for r, 4 for b, and 2 for h in Equation (1).

A=(12×π×42)+(4×2)=20.57

Calculate the mass of given shape:

m=ρA (2)

Substitute 6 for ρ and 20.57 for A in Equation (2).

m=6×20.57=123.398

Consider the f(x) and g(x) as follows:

f(x)=42x2

g(x)=2

Calculate the moment Mx of a lamina with the given density and shape:

Mx=ρab12{[f(x)]2[g(x)]2}dx (3)

Substitute 0 for a, 4 for b, 6 for ρ, 42x2 for f(x), and (2) for g(x) in Equation (3).

Mx=60412[(42x2)2(2)2]dx=6×1204(16x24)dx=304(x2+12)dx (4)

Integrate Equation (4).

Mx=3[x2+1(2+1)+12x]04=3[x33+12x]04=3{[(4)33+12(4)][(0)33+12(0)]}=80

Hence, the moment Mx of a lamina with the given density and shape is 80

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