Question

Two identical copper spheres of radius R and density \(\varrho\) are placed in contact with each other. The gravitational force acting between them is proportional to \(\text{R}^{x}\). The value of \(x\) is

• A. \(7\)
• B. \(6\)
• C. \(4\)
• D. \(2\)

Hint:

Write each mass in terms of R and density, and use centre separation 2R in the gravitational force law.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
The force is \(F=\dfrac{Gm^2}{d^2}\). For touching spheres of radius \(R\), the centres are \(d=2R\) apart.

Step 2: Mass in terms of R
\(m=\dfrac{4}{3}\pi R^3\varrho\)

Step 3: Substitute
\(F=\dfrac{G\left(\frac{4}{3}\pi R^3\varrho\right)^2}{(2R)^2}=\dfrac{4}{9}G\pi^2\varrho^2R^4\)

Step 4: Read the power
So \(F\propto R^4\) and \(x=4\).

Final Answer:
The force is proportional to \(R^4\), so \(x=4\). \[ \boxed{\text{(C) }4} \]