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Question #43905

Difficulty: 8/10 | 🟢 Free

The Question

A small block (treat as a particle) of mass $m = 2 \text{ kg}$ is placed on a rough horizontal surface between two vertical walls separated by a distance $D = 4.0 \text{ m}$ . The block is connected to the left wall by Spring 1 (spring constant $k_1 = 40 \text{ N/m}$ , natural length $L_1 = 1.5 \text{ m}$ ) and to the right wall by Spring 2 (spring constant $k_2 = 60 \text{ N/m}$ , natural length $L_2 = 1.5 \text{ m}$ ). The coefficients of static and kinetic friction between the block and the surface are $\mu_s = 0.9$ and $\mu_k = 0.5$ , respectively. The block is initially held at rest at a distance $x = 1.9 \text{ m}$ from the left wall and then released. Calculate the total heat dissipated by friction from the moment of release until the block comes to a permanent rest. (Take $g = 10 \text{ m/s}^2$ )

Options

A.
$1.0 \text{ J}$
B.
$2.0 \text{ J}$
C.
$4.0 \text{ J}$
D.
$10.0 \text{ J}$

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

Kinetic Friction Transition|Effective Spring Constant|Vector Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly apply the Work-Energy Theorem in a system with a discontinuous friction threshold and pre-stressed springs.

A common trap is confusing the equilibrium position of the kinetic phase ( $F_{net} = 0$ ) with the point of maximum displacement ( $v = 0$ ), leading to a calculated displacement of $0.1 \text{ m}$ and a heat dissipation of $1.0 \text{ J}$ .

Another frequent error is calculating the maximum displacement assuming no friction ( $0.4 \text{ m}$ ) and then erroneously multiplying this distance by the kinetic friction force to find heat ( $4.0 \text{ J}$ ).

Furthermore, students often misidentify springs attached to opposite walls as being in series or subtract their constants ( $k_{eq} = |k_1 - k_2|$ ), which drastically alters the energy balance.

The problem forces a rigorous step-by-step evaluation of the static-to-kinetic transition and the exact non-conservative work done.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Laws of Motion - Class 11 Part 1
Concept	Kinetic Friction Transition
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free