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Question #42393

Difficulty: 8/10 | 🟢 Free

The Question

A rectangular block of width $b$ and height $h$ has a 2D density profile $\rho(x,y) = \rho_0 \left(\frac{x}{b} + \frac{y}{h}\right)$ , where $x \in [0, b]$ and $y \in [0, h]$ are measured from the left and bottom faces respectively. The block is placed on a rough inclined plane with its bottom face resting on the incline and its left face pointing directly up the incline. As the angle of inclination $\theta$ is slowly increased, the block is observed to be on the verge of sliding and toppling simultaneously. What is the coefficient of static friction $\mu$ between the block and the incline?

Options

A.
$\frac{5b}{7h}$
B.
$\frac{7b}{5h}$
C.
$\frac{5b}{12h}$
D.
$\frac{b}{h}$

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

Toppling vs Sliding|Static Friction Threshold|Parallel and Perpendicular Axes

Why This Question Is Tricky

Common Mistakes

This question tests the interplay between torque equilibrium and friction limits, specifically targeting the trap where students assume the normal force acts at the geometric center or that the center of mass of a non-uniform body is at its centroid.

By introducing a 2D density gradient, the problem forces the derivation of the true center of mass.

The critical physical insight is that at the verge of toppling, the normal force shifts entirely to the lower edge, creating a restoring torque condition ( $mg \sin\theta \cdot y_c = mg \cos\theta \cdot x_{pivot}$ ).

Equating this critical toppling angle with the sliding threshold ( $\tan\theta = \mu$ ) synthesizes rigid body mechanics with frictional transition limits.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	System of Particles and Rotational Motion - Class 11 Part 1
Concept	Toppling vs Sliding
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free