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Question #43529

Difficulty: 8/10 | 🟢 Free

The Question

A block of mass $M$ slides at a constant low velocity $v$ over a rough horizontal surface. The macroscopic normal force is $N = Mg$ . Due to microscopic surface roughness, the true area of contact is $A_t = N / P_Y$ , where $P_Y$ is the yield pressure of the softer material. The interface forms adhesive bonds with an area density of $\sigma$ bonds per unit true contact area. Each bond behaves as an ideal linear spring with force constant $k$ and breaks when stretched by a critical distance $x_c$ . After breaking, the bond dissipates its stored elastic energy entirely as heat and immediately reforms at the next lattice site, located at a distance $a$ along the sliding direction. Which of the following expressions correctly represents the coefficient of kinetic friction $\mu_k$ ?

Options

A.
$\frac{\sigma k x_c}{P_Y}$
B.
$\frac{\sigma k x_c^2}{P_Y a}$
C.
$\frac{\sigma k x_c^2}{2 P_Y a}$
D.
$\frac{\sigma k a^2}{2 P_Y x_c}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Interatomic Force Curve|Kinetic Friction Transition|Breaking Stress Concept

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to bridge microscopic interatomic forces with macroscopic non-conservative work.

A common trap is to incorrectly apply Newton's laws by equating the macroscopic friction force directly to the maximum static microscopic force ( $f_k = N_b k x_c$ ), leading to Option A.

This fundamentally misunderstands kinetic friction, which is not a static force balance but an energy dissipation process (Work-Energy Theorem) resulting from the continuous 'stick-slip' shearing and reforming of bonds over a spatial period $a$ .

The correct approach requires integrating the force over the displacement to find the dissipated energy per cycle.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Solids - Class 11 Part 2
Concept	Interatomic Force Curve
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free