Before You Begin

Question #42364

Difficulty: 8/10 | 🟢 Free

The Question

A uniform horizontal circular disc of mass $M$ (in kg) and radius $R$ (in m) is suspended by a vertical wire of torsional constant $K$ (in N $\cdot$ m/rad). Initially, it is undergoing torsional oscillations. At time $t=0$ (in s), the disc passes through its mean position (angular displacement $\theta = 0$ rad) with angular velocity $\omega_0$ (in rad/s). From $t=0$ onwards, sand falls vertically onto the edge of the disc at a constant rate $\mu = \frac{dm}{dt}$ (in kg/s) and sticks to it. Assuming the sand has no initial horizontal velocity, which of the following expressions gives the angular acceleration $\alpha$ (in rad/s $^2$ ) of the disc at any time $t > 0$ in terms of its instantaneous angular displacement $\theta$ and angular velocity $\omega$ ?

Options

A.
$-\frac{K\theta + \mu R^2 \omega}{\frac{1}{2}MR^2 + \mu R^2 t}$
B.
$-\frac{K\theta}{\frac{1}{2}MR^2 + \mu R^2 t}$
C.
$-\frac{K\theta - \mu R^2 \omega}{\frac{1}{2}MR^2 + \mu R^2 t}$
D.
$-\frac{K\theta + \frac{1}{2}\mu R^2 \omega}{\frac{1}{2}MR^2 + \mu R^2 t}$

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

Variable Mass Systems|Angular Momentum conservation|Torsional Oscillations

Why This Question Is Tricky

Common Mistakes

This question tests the application of Newton's Second Law for rotation in variable mass systems.

A common trap is assuming $\tau = I\alpha$ holds universally, leading students to incorrectly write $-K\theta = I(t)\alpha$ .

However, when mass is added at a rate $\mu$ , the correct formulation is $\tau_{ext} = dL/dt = I\alpha + \omega(dI/dt)$ .

The incoming mass must be accelerated to the tangential velocity of the disc, which exerts a dynamic braking torque $-\omega(dI/dt)$ on the system.

This manifests as a velocity-dependent damping term in the torsional oscillator's differential equation.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	System of Particles and Rotational Motion - Class 11 Part 1
Concept	Variable Mass Systems
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free