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Question #42322

Difficulty: 8/10 | 🟢 Free

The Question

A small block of mass $m$ is projected from the lowest point of a fixed, smooth vertical circular track of radius $R$ . The track is immersed in a fluid that exerts a tangential drag force on the block of magnitude $F_d = \frac{m v^2}{2R}$ , where $v$ is the instantaneous speed of the block. What is the minimum initial speed $v_0$ required at the lowest point for the block to just complete the vertical circle? (Assume acceleration due to gravity is $g$ )

Options

A.
$\sqrt{gR(2e^{\pi} + 1)}$
B.
$\sqrt{gR(e^{\pi} + 1)}$
C.
$\sqrt{5gR}$
D.
$\sqrt{gR(2e^{\pi} - 1)}$

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

Potential Energy Reference|Vertical Circular Motion|Kinematics of Vertical Free Fall

Why This Question Is Tricky

Common Mistakes

This question tests the application of the work-energy theorem in the presence of a non-conservative, velocity-dependent drag force during vertical circular motion.

The standard trap is blindly applying mechanical energy conservation ( $v_0 = \sqrt{5gR}$ ), ignoring the path-dependent work done by drag.

Furthermore, it challenges students to correctly set up and solve a differential equation for velocity with respect to angular position, coupling kinematics with Newton's second law, and correctly applying the boundary condition ( $N=0 \implies v^2=gR$ ) at the highest point.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Work, Energy, and Power - Class 11 Part 1
Concept	Potential Energy Reference
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free