Chapter 16.2, Problem 38E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

If a wire with linear density *ρ*(*x*, *y*, *z*) lies along a space curve *C*, its **moments of inertia** about the *x*-, *y*-, and *z*-axes are defined as

*I _{x} =* ∫

*I _{y} =* ∫

*I _{z} =* ∫

Find the moments of inertia for the wire in Exercise 35.

**35.** (a) Write the formulas similar to Equations 4 for the center of mass (
*C* if the wire has density function *ρ*(*x*, *y*, *z*).

(b) Find the center of mass of a wire in the shape of the helix *x =* 2 sin *t*, *y =* 2 cos *t*, *z* = 3*t*, 0 ⩽ *t* ⩽ 2*π*, if the density is a constant *k*.

To determine

**To find:** The moment of inertia for a thin wire about the *x*-, *y*-, and *z*-axes.

Explanation

**Given data:**

The parametric equations of the helix shaped thin wire are given as follows.

The density function

**Formula used:**

Write the expression to find moment of inertia of a thin wire about the *x*-axis.

Here,

Write the expression to find moment of inertia of a thin wire about the *y*-axis.

Write the expression to find moment of inertia of a thin wire about the *z*-axis.

Write the expression for

**Calculation of**
**:**

Substitute

Simplify the expression as follows.

The limits of scalar parameter of curve is

To find the moment of inertial use 0 as lower limit and

**Calculation of moment of inertia about x-axis**

Substitute

Simplify the expression as follows.

Simplify the equation

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