Question

In moving a body of mass \( m \) down a smooth incline of inclination \( \theta \) with velocity \( v \), the power required is (g = acceleration due to gravity):

• A. \( mgv \)
• B. \( (mg \cos \theta) v \)
• C. \( (mg \sin \theta) v \)
• D. \( \frac{mg \sin \theta}{v} \)
• E. \( \frac{mg \cos \theta}{v} \)

Hint:

When calculating power along an incline, always consider the component of the gravitational force along the direction of motion.

Answer 1

The Correct Option is C

We are given that a body of mass \( m \) is moving down a smooth incline with inclination \( \theta \) and velocity \( v \), and we are tasked with finding the power required to move the body. 
Step 1: Understanding the forces involved On an inclined plane, the force due to gravity acting on the body can be split into two components:
- One component acts parallel to the incline: \( mg \sin \theta \).
- The other component acts perpendicular to the incline: \( mg \cos \theta \).
Since the incline is smooth, we assume there is no friction, and only the component of gravitational force parallel to the incline, \( mg \sin \theta \), is responsible for the motion. 
Step 2: Power required Power is the rate at which work is done. The formula for power \( P \) is given by: \[ P = {Force} \times {Velocity} \] Here, the force acting in the direction of motion is \( F = mg \sin \theta \), and the velocity is \( v \). 
Therefore, the power required to move the body is: \[ P = (mg \sin \theta) \times v = (mg \sin \theta) v. \] Thus, the power required is \( (mg \sin \theta) v \), which corresponds to option (C).