Question

A solid cylinder is rolling down on an inclined plane of angle \(\theta\). The coefficient of static friction between the plane and the cylinder is \(\mu_s\). The condition for the cylinder not to slip is

• A. \(\tan\theta \geq 3\mu_s\)
• B. \(\tan\theta>3\mu_s\)
• C. \(\tan\theta \leq 3\mu_s\)
• D. \(\tan\theta<3\mu_s\)

Hint:

For a solid cylinder rolling down an incline, the required friction is $f = Mg\sin\theta/3$, and the no-slip condition is $\tan\theta \leq 3\mu_s$.

Answer 1

The Correct Option is C

Step 1: Underlying Concept:
For rolling without slipping, friction must be sufficient to provide the necessary torque for angular acceleration.
Step 2: Explanation:
Linear acceleration: \(a = \dfrac{2g\sin\theta}{3}\). Friction force: \(f = \dfrac{Mg\sin\theta}{3}\). Normal: \(N = Mg\cos\theta\). Condition \(f \leq \mu_s N\): \[ \frac{Mg\sin\theta}{3} \leq \mu_s Mg\cos\theta \Rightarrow \tan\theta \leq 3\mu_s \]
Step 3: Conclusion:
Condition for no slipping: \(\tan\theta \leq 3\mu_s\).