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Question #43906

Difficulty: 8/10 | 🟢 Free

The Question

A conducting rod of mass $m = 1 \text{ kg}$ and length $L = 1 \text{ m}$ slides on two parallel horizontal rails. A uniform vertical magnetic field $B = 2 \text{ T}$ exists in the region. The rails are connected to a battery of EMF $E = 5 \text{ V}$ and a non-linear semiconductor device in series. The device acts as an ideal diode that only allows current when the battery is discharging (i.e., net EMF drives current in the forward direction), with a forward voltage drop given by $V_s = k I^2$ , where $k = 2 \text{ } \Omega \text{ A}^{-1}$ . The coefficient of kinetic friction between the rod and rails is $\mu_k = 0.4$ , and the coefficient of static friction is $\mu_s = 0.6$ . The rod is given an initial velocity $v_0 = 4 \text{ m/s}$ in the direction of the magnetic force that the battery produces. What is the final steady-state velocity of the rod? (Take $g = 10 \text{ m/s}^2$ )

Options

A.
$0 \text{ m/s}$
B.
$-1.5 \text{ m/s}$
C.
$2.5 \text{ m/s}$
D.
$1.5 \text{ m/s}$

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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

Kinetic Friction Transition|Dynamic Resistance (AC Resistance)|Cell EMF vs Terminal Voltage

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to recognize the physical boundaries of mathematical solutions in non-linear dynamics, specifically the kinetic-to-static friction transition.

Students commonly set the driving magnetic force equal to the kinetic friction to find the terminal velocity, which yields a mathematically valid but physically impossible negative velocity ( $v = -1.5 \text{ m/s}$ ).

They fall into the trap of assuming the system will smoothly reverse direction to reach this state.

However, they fail to account for the fact that when the rod's velocity reaches zero, the governing friction switches from kinetic to static.

Because the magnetic force at rest ( $3.16 \text{ N}$ ) is strictly less than the static friction threshold ( $6 \text{ N}$ ), the rod becomes 'frozen' and cannot accelerate backwards, making the true steady-state velocity zero.

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