Before You Begin

Question #42450

Difficulty: 8/10 | 🟢 Free

The Question

Two fixed concentric uniform spherical shells have masses $M$ and $2M$ , and radii $R$ and $2R$ respectively. An elevator is located in the region between the shells at a radial distance $r$ ( $R < r < 2R$ ) from their common center. The absolute gravitational potential at the elevator's location is $V(r)$ (taking $V \to 0$ at infinity). The elevator is programmed to move such that its radial outward acceleration is exactly $a = \frac{V(r)}{r}$ . A test mass $m$ is suspended from a spring balance inside the elevator. What is the reading of the spring balance when the elevator is at $r = \frac{3R}{2}$ ?

Options

A.
$\frac{2GmM}{3R^2}$
B.
$0$
C.
$\frac{8GmM}{9R^2}$
D.
$\frac{14GmM}{9R^2}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Apparent Weight in Accelerating Elevators|Potential of Concentric Shells|Potential Energy Reference

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between gravitational potential and gravitational field in a non-uniform field generated by concentric shells, while simultaneously applying non-inertial frame concepts.

The primary trap lies in the calculation of the absolute potential $V(r)$ ; students frequently assume the potential inside a shell is zero rather than a non-zero constant ( $-GM_{\text{outer}}/R_{\text{outer}}$ ).

If this constant is omitted, the elevator's programmed acceleration perfectly cancels the local gravitational field, leading to a false reading of zero (weightlessness).

Additionally, students might incorrectly include the outer shell's mass in the local gravitational field calculation or mishandle the vector subtraction of the pseudo-acceleration, leading to the other incorrect options.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Laws of Motion - Class 11 Part 1
Concept	Apparent Weight in Accelerating Elevators
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free