Ch 13: Newton's Theory of Gravity

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 13: Newton's Theory of Gravity

Problem 14

Chapter 13, Problem 14

Suppose we could shrink the earth without changing its mass. At what fraction of its current radius would the free-fall acceleration at the surface be three times its present value?

Verified step by step guidance

Start by recalling the formula for gravitational acceleration at the surface of a planet:

g = \frac{(G M)}{R^{2}}

, where

G

is the gravitational constant,

M

is the mass of the Earth, and

R

is the radius of the Earth.

Since the problem states that the free-fall acceleration becomes three times its current value, let the new acceleration be

g' = 3 g

. Substitute this into the formula for gravitational acceleration:

g' = \frac{(G M)}{{R'}^{2}}

, where

R'

is the new radius.

Divide the equation for

g'

by the equation for

g

to eliminate

G

and

M

\frac{g'}{g} = \frac{R^{2}}{{R'}^{2}}

. Substitute

g' = 3 g

into this equation.

Simplify the equation:

3 = \frac{R^{2}}{{R'}^{2}}

. Take the square root of both sides to solve for the ratio of the radii:

\sqrt{3} = \frac{R}{R'}

Rearrange the equation to find the fraction of the current radius:

R' = \frac{R}{\sqrt{3}}

. This gives the new radius as a fraction of the current radius.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Gravitational Acceleration

Gravitational acceleration is the acceleration experienced by an object due to the gravitational force exerted by a massive body, such as Earth. It is denoted by 'g' and has a standard value of approximately 9.81 m/s² at Earth's surface. This acceleration depends on the mass of the Earth and the distance from its center, following the formula g = G * M / r², where G is the gravitational constant, M is the mass of the Earth, and r is the radius.

Inverse Square Law

The inverse square law states that the strength of a physical quantity (like gravitational force) decreases with the square of the distance from the source. In the context of gravity, if the radius of a planet is reduced while keeping its mass constant, the gravitational acceleration at the surface increases because the distance to the center of mass decreases, leading to a stronger gravitational pull.

Scaling Relationships

Scaling relationships in physics refer to how physical quantities change when the size of an object is altered. In this scenario, if the radius of the Earth is reduced to a fraction of its original size, the gravitational acceleration will increase according to the inverse square of the radius. Understanding these relationships allows us to calculate how changes in size affect gravitational forces and accelerations.

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