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Question #42452

Difficulty: 8/10 | 🟢 Free

The Question

A small block of mass $m$ and charge $+q$ is constrained to move with a constant speed $v$ along a smooth circular groove of radius $R$ cut into the horizontal floor of an elevator. The elevator accelerates upwards with a constant acceleration $a$ . A uniform magnetic field $\vec{B} = B_0 \hat{i} + \sqrt{3}B_0 \hat{j} + 2B_0 \hat{k}$ exists in the region, where $\hat{i}$ and $\hat{j}$ are unit vectors in the horizontal plane and $\hat{k}$ is vertically upwards. Assuming the block never loses contact with the floor, what is the time-averaged apparent weight of the block (the vertical normal force exerted by the floor) over one complete revolution?

Options

A.
m(g+a)
B.
m(g+a) + 2qvB_0
C.
$m(g+a) + \sqrt{2}qvB_0$
D.
$m(g+a) + \frac{4qvB_0}{\pi}$

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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|Magnetic Moment of Orbiting Electron|Pseudo Force Directionality

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to couple non-inertial reference frames with 3D electrodynamics while avoiding mathematical bloat through symmetry.

The 'Lethal Collision' trap lies in the oscillating nature of the vertical Lorentz force.

Students often incorrectly assume they must calculate the maximum apparent weight (Option B), the RMS of the dynamic force (Option C), or the average over a positive half-cycle (Option D).

Furthermore, students might get bogged down calculating the radial normal force or mistakenly assume the vertical magnetic field component $B_z$ contributes to the vertical force.

The key insight is recognizing that $\vec{v} \times \vec{B}$ produces a purely sinusoidal vertical force component that strictly averages to zero over a complete orbit, leaving only the static pseudo-force contribution.

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