Chapter 17, Problem 21RE

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Assume that the earth is a solid sphere of uniform density with mass *M* and radius *R* = 3960 mi. For a particle of mass *m *within the earth at a distance *r* from the earth’s center, the gravitational force attracting the particle to the center is
*G* is the gravitational constant and *M _{r}* is the mass of the earth within the sphere of radius

(a) Show that

(b) Suppose a hole is drilled through the earth along a diameter. Show that if a particle of mass *m* is dropped from rest at the surface, into the hole, then the distance *y* = *y*(*t*) of the particle from the center of the earth at time *t* is given by *y"*(*t*) = -*k*^{2}*y*(*t*)

where *k*^{2} = *GM/R*^{3} = *g*/*R.*

(c) Conclude from part (b) that the particle undergoes simple harmonic motion. Find the period *T.*

(d) With what speed does the particle pass through the center of the earth?

(a)

To determine

**To show:** The expression

Explanation

**Given data:**

Consider the radius of earth *R* as 3960 mi.

Calculate the density

Consider the expression for the volume of earth.

Substitute *M* for mass of earth, and

If
*r* of the center, then,

Consider the expression for the mass of earth within the sphere of radius *r*,

Substitute

(b)

To determine

**To show:** The expression

(c)

To determine

**To find:** the period *T* from the expression

(d)

To determine

**To find:** the speed of the particle passing through the center of the earth.

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