Before You Begin

Question #42489

Difficulty: 8/10 | 🟢 Free

The Question

A fixed horizontal cylinder of radius $R$ is centered at the origin $O$ . A massless rigid tube is pivoted at $O$ and can rotate freely in the vertical plane. A point mass $m$ slides without friction inside the tube. An inextensible massless string is attached to the mass $m$ , runs straight through the tube to a small smooth ring fixed to the tube at a distance $R$ from $O$ (which lies on the surface of the cylinder), and then wraps around the fixed cylinder. The other end of the string is fixed to the lowest point of the cylinder. As the tube rotates by an angle $\theta$ from the downward vertical, the string wraps around the cylinder, pulling the mass $m$ inwards such that its distance from $O$ is $r(\theta) = L - R\theta$ (where $L$ is a constant length). The system is released from rest at an angle $\theta = \theta_0$ (where $0 < \theta_0 < \pi/2$ ). What is the magnitude of the initial angular acceleration $\alpha$ of the tube?

Options

A.
$\frac{g [R \cos \theta_0 + (L - R\theta_0) \sin \theta_0]}{R^2 + (L - R\theta_0)^2}$
B.
$\frac{g \sin \theta_0}{L - R\theta_0}$
C.
$\frac{g (L - R\theta_0) \sin \theta_0}{R^2 + (L - R\theta_0)^2}$
D.
$\frac{g [R \cos \theta_0 + (L - R\theta_0) \sin \theta_0]}{(L - R\theta_0)^2}$

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

Mechanical Energy Conservation|Compound Pendulum Period|Vector Cross-Product

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to analyze a system with a variable moment of inertia and holonomic constraints.

The lethal trap is blindly applying $\tau = I\alpha$ using only the gravitational torque and the instantaneous moment of inertia $I = mr^2$ .

This ignores two critical factors: the radial kinetic energy which modifies the effective moment of inertia to $m(r^2 + R^2)$ , and the external constraint torque from the fixed cylinder on the wrapping string.

Formulating the Lagrangian $\mathcal{L} = T - U$ naturally accounts for the constraint work and the correct generalized force, revealing an additional restoring torque term $mgR\cos\theta_0$ caused by the string's wrapping mechanism.

Quick Facts About This Question

Question Information

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