Chapter 3.4, Problem 12P

(a)

Program Plan Intro

Program Description: The purpose of the problem is to show that for a particle of mass $m$ within the earth at distance $r$ from the center of the earth, the gravitational force attracting $m$ toward the centeris $F_r=\frac{-GMmr}{R^3}$ such that $M$ is the mass of the earth and $R$ is the radius of the earth equal to $3960\text{ mi}$ if the gravitational force attracting $m$ toward the center is given by $F_r=\frac{-G{M}_{r}m}{r^2}$ such that $M_r$ is the mass of the earth within a sphere of radius $r$ .

(b)

Program Plan Intro

Program Description: The purpose of the problem is to conclude from part (a) and Newton’s second law that $r\text{'}\text{'}\left(t\right)=-k^{2}r\left(t\right)$ such that $k^2=\frac{GM}{R^3}=\frac{g}{R}$ if a particle of mass $m$ is dropped at time $t=0$ into a small hole drilled straight through the center of the earth with initial speed zero and $r\left(t\right)$ is the distance of the mass from the center of the earth at time $t$ .

(c)

Program Plan Intro

Program Description: The purpose of the problem is to conclude from part (b) that a particle of mass $m$ is dropped at time $t=0$ into a small hole drilled straight through the center of the earth with initial speed zero and $r\left(t\right)$ is the distance of the mass from the center of the earth at time $t$ undergoes simple harmonic motion back and forth between the ends of the hole with a period of about $84\text{ min}\text{.}$

(d)

Program Plan Intro

Program Description: The purpose of the problem is to find the period of a satellite that skims the surface of the earth

by use of above parts and compare the result with the period found in part (c).

(e)

Program Plan Intro

Program Description: The purpose of the problem is to find the speed of the particle in miles per hour on passing through the center of the earth.

(f)

Program Plan Intro

Program Description: The purpose of the problem is to find orbital velocity of a satellite that just skims the surface of the earth, compare the result with part (e)