Question

Two spheres exert a gravitational attraction on each other. Which of the following changes to this system would reduce the gravitational attraction by one half?

Answer 1

- Decreasing one of the masses to 1/2 of its original value

- Increasing the distance by a factor of $\sqrt{2}$

Explanation:

There are no options provided, however we can still answer the question.

In fact, the magnitude of the gravitational force between two objects is given by  the equation:

$F=G\frac{m_1 m_2}{r^2}$

where

$G=6.67\cdot 10^{-11} m^3 kg^{-1}s^{-2}$ is the gravitational constant

m1, m2 are the masses of the two objects

r is the separation between them

We notice that:

• The gravitational force is proportional to the product between the masses
• The gravitational force is inversely proportional to the square of the distance between the masses

Therefore, in order to reduce the gravitational attraction by one half, we can do one of the following changes:

- Decreasing one of the masses to 1/2 of its original value: for example, if $m_1'=\frac{1}{2}m_1$, the gravitational force becomes

$F'=G\frac{m_1' m_2}{r^2}=G\frac{\frac{1}{2}m_1m_2}{r^2}=\frac{1}{2}(G\frac{m_1m_2}{r^2})=\frac{1}{2}F$

- Increasing the distance by a factor of $\sqrt{2}$: in fact, if $r'=\sqrt{2}r$, the gravitational force becomes

$F'=G\frac{m_1 m_2}{r'^2}=G\frac{m_1m_2}{(\sqrt{2}r)^2}=\frac{1}{2}(G\frac{m_1m_2}{r^2})=\frac{1}{2}F$

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