Question

How does the force of gravitation between two objects change when the distance between them is reduced to half ?

• A. 2 times
• B. 4 times
• C. becomes half
• D. remains same

Hint:

Gravitational force \(F \propto \frac{1}{r^2}\) (inversely proportional to the square of the distance). 1. Let original distance = \(r\). Original force \(F_{old} = \frac{k}{r^2}\) (where \(k = Gm_1m_2\)). 2. New distance = \(r/2\). 3. New force \(F_{new} = \frac{k}{(r/2)^2} = \frac{k}{r^2/4} = \frac{4k}{r^2}\). 4. Compare: \(F_{new} = 4 \times \frac{k}{r^2} = 4 \times F_{old}\). So, the force becomes 4 times the original force. If distance is halved, force is quadrupled. If distance is doubled, force becomes 1/4.

Answer 1

The Correct Option is B

Concept: Newton's Law of Universal Gravitation states that the gravitational force (\(F\)) between two objects of masses \(m_1\) and \(m_2\), separated by a distance \(r\) between their centers, is given by: \[ F = G \frac{m_1 m_2}{r^2} \] where \(G\) is the gravitational constant. This formula shows that the force of gravitation is inversely proportional to the square of the distance between the objects (\(F \propto \frac{1}{r^2}\)). Step 1: Write the initial force Let the initial distance between the two objects be \(r_1\). The initial gravitational force \(F_1\) is: \[ F_1 = G \frac{m_1 m_2}{r_1^2} \] Step 2: Define the new distance The distance between them is reduced to half. Let the new distance be \(r_2\). So, \(r_2 = \frac{r_1}{2}\). Step 3: Calculate the new force (\(F_2\)) with the new distance The new gravitational force \(F_2\) with distance \(r_2\) is: \[ F_2 = G \frac{m_1 m_2}{r_2^2} \] Substitute \(r_2 = \frac{r_1}{2}\): \[ F_2 = G \frac{m_1 m_2}{\left(\frac{r_1}{2}\right)^2} \] \[ F_2 = G \frac{m_1 m_2}{\frac{r_1^2}{4}} \] To simplify the fraction in the denominator, multiply the numerator by the reciprocal of \(\frac{r_1^2}{4}\), which is \(\frac{4}{r_1^2}\): \[ F_2 = G \frac{m_1 m_2}{1} \times \frac{4}{r_1^2} \] \[ F_2 = 4 \left( G \frac{m_1 m_2}{r_1^2} \right) \] Step 4: Relate the new force (\(F_2\)) to the initial force (\(F_1\)) We know from Step 1 that \(F_1 = G \frac{m_1 m_2}{r_1^2}\). Substituting this into the expression for \(F_2\): \[ F_2 = 4 F_1 \] This means the new force \(F_2\) is 4 times the initial force \(F_1\). So, when the distance is reduced to half, the force of gravitation becomes 4 times stronger. This matches option (2).