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Question #43852

Difficulty: 8/10 | 🟢 Free

The Question

A simple pendulum consists of a small bob of mass $m$ attached to a rigid, massless rod of length $L$ . The other end of the rod is rigidly attached to a cylindrical shaft of radius $r$ ( $r \ll L$ ) and negligible mass. The shaft rotates in a fixed, horizontal dry bearing with a coefficient of kinetic friction $\mu$ . The pendulum is swinging upwards, and at the instant it makes an angle $\theta$ with the downward vertical, its angular speed is $\omega$ . Assume that the normal force exerted by the bearing on the shaft is equal to the radial tension in the rod. Which of the following expressions correctly represents the magnitude of the angular acceleration $\alpha$ of the pendulum at this instant?

Options

A.
$\frac{g}{L} \sin\theta + \mu \frac{r}{L^2} \left( g \cos\theta + L \omega^2 \right)$
B.
$\frac{g}{L} \sin\theta + \mu \frac{r}{L^2} g \cos\theta$
C.
$\frac{g}{L} \sin\theta - \mu \frac{r}{L^2} \left( g \cos\theta + L \omega^2 \right)$
D.
$\frac{g}{L} \sin\theta + \mu \frac{r}{L} \omega^2$

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

Physical Pendulum Inertia|Kinetic Friction Transition|Differential Equations

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to recognize that the damping torque in a physical system with dry friction is not necessarily constant.

A common trap is assuming the normal force on the pivot is simply the static weight component ( $mg \cos\theta$ ), ignoring the dynamic centrifugal term ( $mL\omega^2$ ) required for circular motion.

By coupling the radial equation of motion to the rotational equation of motion, the problem highlights a non-linear feedback mechanism where the angular velocity directly influences the instantaneous frictional damping.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Oscillations - Class 11 Part 2
Concept	Physical Pendulum Inertia
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free