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Question #43477

Difficulty: 8/10 | 🟢 Free

The Question

Block A of mass $m$ is attached to a linear spring of force constant $k$ , with the other end of the spring fixed to a rigid wall. Block A rests on a smooth horizontal surface. Block B, also of mass $m$ , is connected to Block A by a light, inextensible slack string of length $L$ . Initially, the spring is relaxed and Block A is at rest. Block B is projected away from Block A with a horizontal velocity $v_0$ along the axis of the spring. At time $t = t_0$ , the string becomes taut. What is the magnitude of the average tension in the string over the time interval from just before the string becomes taut ( $t = t_0^-$ ) until the two-block system comes to instantaneous rest for the first time?

Options

A.
$\frac{v_0}{\pi} \sqrt{2km}$
B.
$\frac{v_0}{2\pi} \sqrt{2km}$
C.
$\frac{2v_0}{\pi} \sqrt{km}$
D.
$\frac{v_0}{\pi} \sqrt{km}$

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

Average vs. Instantaneous Calculus|SHM Definition via Differential Equation|Differential Calculus

Why This Question Is Tricky

Common Mistakes

This problem tests the student's ability to reconcile impulsive dynamics with the continuous differential equations of Simple Harmonic Motion (SHM).

The primary trap is calculating the average tension by only integrating the restorative SHM phase ( $\int T_{SHM} dt = m v_0 / 2$ ) while ignoring the Dirac-delta nature of the impulsive tension at $t=t_0$ ( $\int T_{imp} dt = m v_0 / 2$ ).

Students who fall for this trap will select Option B.

By applying the Impulse-Momentum theorem directly to Block B over the entire interval, students can elegantly bypass complex piecewise integration, demonstrating a deeper mastery of average vs. instantaneous calculus in physics.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Straight Line - Class 11 Part 1
Concept	Average vs. Instantaneous Calculus
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free