Question

On a rough inclined plane (with angle of incline $\alpha$) lies a body to which a light, inextensible thread is tied (see Fig.). The end of the thread is passed through a small hole in the plane. At the initial moment, the thread is horizontal. The thread is pulled out very slowly, and by the time it reaches the hole, it describes half a circle. Which of the following options is correct?

• A. The angle of incline is less than the angle of friction.
• B. The angle of incline is equal to the angle of friction.
• C. The angle of incline is more than the angle of friction.
• D. The angle of incline and the angle of friction could not be compared.

Answer 1

Answer: (C)

The angle of incline is more than the angle of friction.

Answer 2

If the angle of incline $\alpha$ is greater than the angle of friction $\phi$ ($\alpha > \phi$), the body has a tendency to slide down the incline due to gravity overcoming static friction. In this scenario, the thread is pulled out, implying that the thread is being manipulated to control this downward motion. The phrase "pulled out very slowly" suggests a controlled movement, which is consistent with managing a sliding body. If $\alpha \le \phi$, the body would not slide down on its own, and pulling the thread would be the primary action to initiate motion. However, the problem statement allows for the interpretation that the thread is controlling an existing tendency to slide, making $\alpha > \phi$ the most appropriate condition.