Question

A force F acting on a body depends on its displacement S as \(F \propto S^{-1/3}\). The power delivered by F will depend on displacement as

• A. \(S^{2/3}\)
• B. \(S^{-5/3}\)
• C. \(S^{1/2}\)
• D. \(S^0\)

Hint:

When \(F \propto S^{-1/3}\), \(v \propto S^{1/3}\), so power is constant.

Answer 1

The Correct Option is D

To solve this problem, we need to determine how the power delivered by a force \( F \) depends on the displacement \( S \), given that the force is proportional to the displacement raised to the power of \(-1/3\). Let's follow the steps below:

1. Understanding the Relationship Between Force and Displacement:
   • The problem states that \( F \propto S^{-1/3} \). This can be expressed as: \(F = k S^{-1/3}\), where \( k \) is a constant of proportionality.
2. Relating Power to Force and Velocity:
   • Power \( P \) is defined as the rate at which work is done. It can also be calculated using the formula: \(P = F \cdot v\), where \( v \) is the velocity of the object.
3. Determining Velocity in Terms of Displacement:
   • Velocity \( v \) is the rate of change of displacement with respect to time: \(v = \frac{dS}{dt}\).
4. Substituting Force and Velocity in the Power Expression:
   • Substitute the expression for \( F \) (from step 1) and velocity (from step 3) into the power formula: \(P = (k S^{-1/3}) \cdot \frac{dS}{dt}\).
5. Simplifying the Expression:
   • We recognize that the velocity \( v \) is essentially the rate of change of displacement, so \(v \cdot \frac{1}{v} = 1\).
   • Hence, the power depends on the displacement  S a

Therefore, the correct answer is: \(S^0\), indicating that the power delivered by the force is constant and does not depend on the displacement.