   Chapter 17, Problem 21RE

Chapter
Section
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 F r = − G M r m r 2 where G is the gravitational constant and Mr is the mass of the earth within the sphere of radius r.(a) Show that F r = − G M m R 3 r . (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) = -k2y(t)where k2 = GM/R3 = 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 Fr=GMmR3r

Explanation

Given data:

Consider the radius of earth R as 3960 mi.

Calculate the density ρ of earth by considering the earth is a solid of uniform density.

ρ=massofearthvolumeofearth (1)

Consider the expression for the volume of earth.

V=43πR3

Substitute M for mass of earth, and 43πR3 for volume of earth in equation (1),

ρ=M43πR3

If Vr is the volume of the portion of earth which lies within a distance r of the center, then,

Vr=43πr3

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

Mr=ρVr

Substitute M43πR3 for ρ and 43πr3 for Vr ,

Mr

(b)

To determine

To show: The expression y(t)=k2y(t) .

(c)

To determine

To find: the period T from the expression y(t)+k2y(t)=0 .

(d)

To determine

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

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