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Chapter 8: Problem 13

To what fraction of its current radius would Earth have to shrink (with no change in mass) for the gravitational acceleration at its surface to triple?

Short Answer

Expert verified
The Earth's radius would need to shrink to about 57.7% of its current size to triple the gravitational acceleration at its surface, assuming no change in Earth's mass.

Step by step solution

01

Understand the Gravitational Force Equation

The gravitational force equation is represented as \[ F = G \cdot \frac{m_1 \cdot m_2}{r^2} \], where G is the gravitational constant, \(m_1\) and \(m_2\) are the masses of the two objects and r is the distance between the centers of the two objects. Here, \(m_1\) can be the mass of the Earth, \(m_2\) the mass of another object (like an object on Earth's surface), and r would be the Earth's radius.
02

Set Up the Problem

We are asked how much the Earth's radius needs to be reduced in order for the gravitational acceleration to triple. That means, the gravitational acceleration will become \(3g\), where g is the current gravitational acceleration on Earth's surface. It can be stated as \[ 3g = G \cdot \frac{M_e}{r_{new}^2} \], where \(M_e\) is Earth's mass and \(r_{new}\) is the 'new' radius.
03

Simplify the Equation

We know that Earth's gravitational acceleration g can be represented as \(g = G \cdot \frac{M_e}{r_{old}^2}\), where \(r_{old}\) is the current radius. Substitute the value of \(g\) in the equation from step 2 and get \[ 3 \cdot G \cdot \frac{M_e}{r_{old}^2} = G \cdot \frac{M_e}{r_{new}^2} \].
04

Solve for the New Radius

Solving the equation from step 3 for \(r_{new}\) yields \[ r_{new} = r_{old} \cdot \sqrt[2]{\frac{1}{3}} \]. The denominator inside the square root signifies that the radius needs to be reduced by a factor of \(\sqrt[2]{\frac{1}{3}}\) or approximately 0.577 to make the gravitational acceleration triple. That is to say, the Earth's radius would need to shrink to about 57.7% of its current size to triple the gravitational acceleration at its surface, assuming no change in Earth's mass.

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Most popular questions from this chapter

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