Question

Consider two identical metallic spheres of radius $R$ each having charge $Q$ and mass $m$. Their centers have an initial separation of $4R$. Both the spheres are given an initial speed of $u$ towards each other. The minimum value of $u$, so that they can just touch each other is :
(Take $k = \frac{1}{4 \pi \epsilon_0}$ and assume $kQ^2>Gm^2$ where $G$ is the Gravitational constant)

• A. \(\sqrt{\frac{kQ^2}{4mR} \left( 1 - \frac{Gm^2}{kQ^2} \right)}\)
• B. \(\sqrt{\frac{kQ^2}{2mR} \left( 1 - \frac{Gm^2}{kQ^2} \right)}\)
• C. \(\sqrt{\frac{kQ^2}{2mR} \left( 1 - \frac{Gm^2}{2kQ^2} \right)}\)
• D. \(\sqrt{\frac{kQ^2}{4mR} \left( 1 + \frac{Gm^2}{kQ^2} \right)}\)

Hint:

When two identical objects move toward each other, remember that the total kinetic energy is $m u^2$. For the "just touch" condition, the final separation is twice the radius.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The spheres move under the influence of two central forces: the electrostatic repulsive force and the gravitational attractive force. Since the net potential energy of the system depends only on the distance between the spheres, mechanical energy is conserved. To "just touch," the spheres must reach a center-to-center distance of $2R$ with a final relative velocity of zero.

Step 2: Key Formula or Approach:
We use the Principle of Conservation of Mechanical Energy:
\[ K_i + U_i = K_f + U_f \]
Where potential energy \( U = U_e + U_g = \frac{kQ^2}{r} - \frac{Gm^2}{r} = \frac{1}{r}(kQ^2 - Gm^2) \).

Step 3: Detailed Explanation:
Initial State: Separation \( r_i = 4R \), each sphere has speed $u$.
Total initial kinetic energy \( K_i = \frac{1}{2} m u^2 + \frac{1}{2} m u^2 = m u^2 \).
Initial potential energy \( U_i = \frac{kQ^2 - Gm^2}{4R} \).
Final State: Separation \( r_f = 2R \), speeds are zero (minimum $u$).
Total final kinetic energy \( K_f = 0 \).
Final potential energy \( U_f = \frac{kQ^2 - Gm^2}{2R} \).
Applying conservation of energy:
\[ m u^2 + \frac{kQ^2 - Gm^2}{4R} = 0 + \frac{kQ^2 - Gm^2}{2R} \]
\[ m u^2 = \frac{kQ^2 - Gm^2}{2R} - \frac{kQ^2 - Gm^2}{4R} \]
\[ m u^2 = \frac{kQ^2 - Gm^2}{4R} \]
\[ u^2 = \frac{kQ^2}{4mR} \left( 1 - \frac{Gm^2}{kQ^2} \right) \]
\[ u = \sqrt{\frac{kQ^2}{4mR} \left( 1 - \frac{Gm^2}{kQ^2} \right)} \]

Step 4: Final Answer:
The minimum speed required is \(\sqrt{\frac{kQ^2}{4mR} \left( 1 - \frac{Gm^2}{kQ^2} \right)}\).