Chapter 16.2, Problem 33E

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8th Edition

James Stewart

ISBN: 9781305266643

Textbook Problem

A thin wire is bent into the shape of a semicircle *x*^{2} + *y*^{2} = 4, *x* ⩾ 0. If the linear density is a constant *k*, find the mass and center of mass of the wire.

To determine

**To find:** The mass of a thin wire which bent into the shape of a semicircle

Explanation

**Given data:**

The mass of a thin wire which bent into the shape of a semicircle

The linear density is a constant

**Formula used:**

Write the expression to find the mass of a thin wire which is in the shape of a curve

Here,

Write the expression to find

Write the expression to find the *x*-coordinate of a center of mass of the thin wire.

Here,

Write the expression to find the *y*-coordinate of a center of mass of the thin wire.

Write the equation of semi circle as follows.

Consider the parametric equations such that the considered parameters must satisfy the equation of the circle

Consider the parametric equations of the curve as follows.

The considered parametric equations and the limits are satisfied if the parameters substitute in the equation of circle

As the linear density is a constant

**Calculation of**
**:**

Substitute

Simplify the expression as follows.

**Calculation of mass**
**:**

Substitute

Thus, the mass of a thin wire which bent into the shape of a semicircle

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