   Chapter 16.2, Problem 35E

Chapter
Section
Textbook Problem

(a) Write the formulas similar to Equations 4 for the center of mass ( x ¯ ,   y ¯ ,   z ¯ ) of a thin wire in the shape of a space curve 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 = 3t, 0 ⩽ t ⩽ 2π, if the density is a constant k.

(a)

To determine

To write: The formulae for the center of mass of a thin wire which is in the shape of a space curve C .

Explanation

Given data:

The thin wire is in the shape of a space curve C and the density function of the wire is ρ(x,y,z) .

Consider a thin wire which is in the shape of a space curve C and the density function of the wire is ρ(x,y,z) at the point (x,y,z) .

The center of mass of the thin wire is located at the point (x¯,y¯,z¯) .

The formulae for the x-, y-, and z-coordinates of center of mass of the thin wire are written as follows.

Formula for the x-coordinate of center of mass of the thin wire:

x¯=1mCxρ(x,y,z)ds (1)

Here,

m is the mass of the thin wire,

x is the x-coordinate of the point (x,y,z) ,

ρ(x,y,z) is the density function of the wire, and

ds is the change in surface.

Write the expression for the mass of the thin wire.

m=Cρ(x,y,z)ds (2)

Write the expression for ds

(b)

To determine

To find: The center of mass of a thin wire which is in the shape of a helix.

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