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Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

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BuyFindarrow_forward

Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

Finding the Radius of Gyration About Each Axis in Exercises 29-34, verify the given moment(s) of inertia and find x ¯ ¯ and y ¯ ¯ . Assume that each lamina has a density of ρ = 1 gramper square centimeter. (These regions are common shapes use din engineering.)

Rectangle

images

To determine

To calculate: The values of x¯¯ and y¯¯ and also, verify the moment of inertia of the provided lamina.

images

Explanation

Given:

The given lamina is shown in the figure below:

Assume the density of lamina as ρ=1.

Formula Used:

The moment of inertia about x – axis is determined by the formula, Ix=0b0hy2dydx.

The moment of inertia about y – axis is determined by the formula, Iy=0b0hx2dydx.

Calculation:

First, calculate the mass of the provided lamina as follows,

m=0b0hdydx=0b[y]0hdx=0bhdx=[hx]0b

This gives,

m=bh

The moment of inertia about x-axis is calculated as shown below:

Ix=0b0hy2dydx=0b[y33]0hdx=0bh33dx=h33[x]0b

Then,

Ix=bh33

This shows moment of inertia about x-axis is equal to the provided value moment of inertia

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