Question

In the adjoining figure, if the pulley and the inclined plane are frictionless, then upward acceleration of the mass $ m $ will be

• A. zero
• B. \( \frac{g}{4} \)
• C. \( \frac{g}{3} \)
• D. \( \frac{g}{2} \)

Hint:

For frictionless systems involving inclined planes and pulleys, break the forces into components and apply Newton's second law to solve for the acceleration.

Answer 1

The Correct Option is C

Step 1: Understanding the forces.
Since the pulley and the inclined plane are frictionless, we focus on the forces acting on the system. The force on the mass \( m \) is due to gravity, and the tension in the rope affects its upward acceleration.
Step 2: Force analysis.
The component of the gravitational force acting down the inclined plane is \( mg \sin(30^\circ) \). The tension in the rope \( T \) is responsible for the upward motion of mass \( m \). Since the system is massless and frictionless, the net force on the system will be balanced, and we can apply Newton's second law to solve for the acceleration. After calculating, we find that the acceleration of the mass \( m \) is \( \frac{g}{3} \). Thus, the correct answer is
(C) \( \frac{g}{3} \)
.