Question

complete the sentences with the best statement.
doubling the distance between two
objects
halving the distance between two
objects
tripling the distance between two
objects
reducing the distance between two
objects by one third
drag & drop the answer
reduces the gravitational force between them
doubles the gravitational force between them
brings the gravitational force between them
increases the force between them by a factor of
four.
increases the gravitational force between them by a
factor of four.

Explanation:

Step1: Recall the gravitational force formula

The gravitational force formula is \(F = \frac{Gm_1m_2}{r^{2}}\), where \(G\) is the gravitational constant, \(m_1\) and \(m_2\) are the masses of the two objects, and \(r\) is the distance between them.

Step2: Analyze doubling the distance

If \(r\) becomes \(2r\), then \(F'=\frac{Gm_1m_2}{(2r)^{2}}=\frac{Gm_1m_2}{4r^{2}}=\frac{1}{4}F\). So doubling the distance reduces the gravitational force between them to a quarter.

Step3: Analyze halving the distance

If \(r\) becomes \(\frac{r}{2}\), then \(F'=\frac{Gm_1m_2}{(\frac{r}{2})^{2}}=\frac{4Gm_1m_2}{r^{2}} = 4F\). So halving the distance increases the gravitational force between them by a factor of four.

Step4: Analyze tripling the distance

If \(r\) becomes \(3r\), then \(F'=\frac{Gm_1m_2}{(3r)^{2}}=\frac{Gm_1m_2}{9r^{2}}=\frac{1}{9}F\). So tripling the distance reduces the gravitational force between them to a ninth.

Step5: Analyze reducing the distance by one - third

If \(r\) becomes \(\frac{2r}{3}\) (since reducing by one - third means the new distance is \(r-\frac{1}{3}r=\frac{2r}{3}\)), then \(F'=\frac{Gm_1m_2}{(\frac{2r}{3})^{2}}=\frac{9Gm_1m_2}{4r^{2}}=\frac{9}{4}F\). So reducing the distance between two objects by one - third increases the gravitational force between them by a factor of \(\frac{9}{4}\).

Answer:

• Doubling the distance between two objects: reduces the gravitational force between them to a quarter.
• Halving the distance between two objects: increases the gravitational force between them by a factor of four.
• Tripling the distance between two objects: reduces the gravitational force between them to a ninth.
• Reducing the distance between two objects by one - third: increases the gravitational force between them by a factor of \(\frac{9}{4}\).