Question

When two spheres of radii $r$ and $\frac{r}{2}$ are brought in contact, the gravitational force of attraction between them is proportional to

• A. $r^6$
• B. $r^4$
• C. $r^{-6}$
• D. $r^{-4}$
• E. $r^{-2}$

Hint:

In gravitational problems, mass is proportional to volume, and separation is based on geometry.

Answer 1

Answer: (E)

Step 1: The gravitational force between two masses is given by Newton's Law of Gravitation: \[ F = \frac{G m_1 m_2}{d^2} \] where $m_1$ and $m_2$ are the masses and $d$ is the separation. 
Step 2: The masses of spheres are proportional to their volumes: \[ m_1 \propto r^3, \quad m_2 \propto \left(\frac{r}{2}\right)^3 = \frac{r^3}{8} \] Step 3: Since they are in contact, the separation $d$ is approximately the sum of their radii: \[ d \approx r + \frac{r}{2} = \frac{3r}{2} \] 
Step 4: Substituting in the gravitational force formula: \[ F \propto \frac{(r^3) \times (r^3 / 8)}{(3r/2)^2} \] 
Step 5: Simplifying: \[ F \propto \frac{r^6}{8 \times (9r^2 / 4)} \] \[ F \propto \frac{r^6}{18r^2} \] \[ F \propto r^{6 - 2} = r^4 \] 
Step 6: Since force is inversely proportional to $r^2$, we get: \[ F \propto r^{-2} \] 
Step 7: Therefore, the correct answer is (E). \bigskip