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Question #42803

Difficulty: 8/10 | 🟢 Free

The Question

A solid conducting sphere of uncharged rest mass $m_0$ and radius $R$ is given a total charge $Q$ . The sphere is placed on a rough horizontal platform and attached to one end of a relaxed ideal spring of force constant $k$ . The other end of the spring is fixed to a vertical wall attached to the platform. The coefficient of kinetic friction between the sphere and the platform is $\mu_k$ . The platform is suddenly given a constant horizontal acceleration $a$ ( $a > \mu_k g$ ) in a direction such that the sphere moves away from the wall, stretching the spring. Assume the sphere is constrained to slide without rolling. In the reference frame of the platform, what is the total work done by the pseudo-force on the sphere from the instant the platform starts accelerating until the sphere reaches its maximum displacement relative to the platform for the first time? (Here $c$ is the speed of light in vacuum and $\epsilon_0$ is the permittivity of free space).

Options

A.
$\frac{2 a (a - \mu_k g)}{k} \left( m_0 + \frac{Q^2}{8 \pi \epsilon_0 R c^2} \right)^2$
B.
$\frac{2 a (a - \mu_k g)}{k} m_0^2$
C.
$\frac{2 (a - \mu_k g)^2}{k} \left( m_0 + \frac{Q^2}{8 \pi \epsilon_0 R c^2} \right)^2$
D.
$\frac{2 a (a - \mu_k g)}{k} \left( m_0 + \frac{3Q^2}{20 \pi \epsilon_0 R c^2} \right)^2$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Self-Energy of Charged Sphere|Static Friction Threshold|Frame of Reference Inertial vs Non-Inertial

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of relativistic mass-energy equivalence with classical non-inertial mechanics and the work-energy theorem.

The primary trap lies in the definition of the inertial mass.

Students commonly use only the rest mass $m_0$ to calculate the pseudo-force and normal force, ignoring the mass equivalent of the electrostatic self-energy ( $U/c^2$ ) stored in the field of the highly charged sphere.

By failing to account for this 'electromagnetic mass', they incorrectly evaluate both the pseudo-force doing the work and the frictional force opposing it.

Furthermore, the quadratic dependence on the effective mass in the final work expression heavily penalizes this omission, distinguishing those who deeply understand the equivalence principle from those applying rote classical formulas.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Self-Energy of Charged Sphere
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free