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Question #42948

Difficulty: 8/10 | 🟢 Free

The Question

Two isolated point masses, $m_1 = 2M$ and $m_2 = 3M$ , are initially held at rest in free space at a separation $d = 4R$ . They are released simultaneously and move towards each other under their mutual gravitational attraction. Consider an imaginary spherical surface $S$ of radius $R$ centered on $m_2$ and translating rigidly with it. What is the kinetic energy of $m_1$ as measured by a non-rotating observer fixed on $m_2$ , at the exact instant $m_1$ crosses the surface $S$ ? (Here, $G$ is the universal gravitational constant)

Options

A.
$\frac{15 GM^2}{2R}$
B.
$\frac{9 GM^2}{2R}$
C.
$\frac{27 GM^2}{10R}$
D.
$\frac{9 GM^2}{5R}$

Check Your Answer

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

Equipotential Surface Geometry|Gravitational Potential Energy (System)|Reduced Mass Effect

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to navigate between inertial and non-inertial reference frames while applying energy conservation.

The primary trap lies in assuming that the work done on $m_1$ in the frame of $m_2$ can be calculated using only the standard gravitational potential $V(r) = -Gm_2/r$ .

If a student does this, they will calculate the work as $m_1 \Delta V = 4.5 GM^2/R$ (Option B), completely ignoring the pseudo-force acting on $m_1$ due to the acceleration of $m_2$ .

The correct physical approach requires recognizing that in the non-inertial frame of $m_2$ , the effective potential is $V_{eff} = -G(m_1+m_2)/r$ , or more elegantly, shifting to the Center-of-Mass frame to find the relative velocity using the reduced mass $\mu$ , and then calculating the kinetic energy of $m_1$ using its actual mass ( $2M$ ) and the relative velocity.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Equipotential Surface Geometry
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free