   Chapter 13, Problem 73AP

Chapter
Section
Textbook Problem

Assume a hole is drilled through the center of the Earth. It can be shown that an object of mass m at a distance r from the center of the Earth is pulled toward the center only by the material in the shaded portion of Figure P13.73. Assume Earth has a uniform density ρ. Write down Newton’s law of gravitation for an object at a distance r from the center of the Earth and show that the force on it is of the form of Hooke’s law, F = −kr, with an effective force constant of k = ( 4 3 ) π ρ G m , where G is the gravitational constant. Figure P13.73

To determine
The gravitational force on an object which is at a distance r from the center of the earth is of the form of Hooke’s law.

Explanation

Given info: The mass of the object is m . The uniform density of earth is ρ . The distance of the object from the center of earth is r .

The Newton’s law of gravitation is,

F=GmMr2 (1)

• F is the gravitational force
• G is the gravitational constant
• m is the mass of the object
• M is the mass of earth beneath the object
• r is the distance of object from the center of earth

The mass of earth beneath the object can be found out as,

M=ρ(43πr3) (2)

From (1) and (2),

F=G

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