A particle moves along a curve of unknown shape but magnitude of force `F` is constant and always acts along tangent to the curve.Then

Options

A. `F` may be conservative
B. `F` must be conservative
C. `F` may be non conservative
D. `F` must be non-conservative

Explanation

To solve the question, we need to analyze the motion of a particle moving along a curve under the influence of a constant force \( F \) that always acts along the tangent to the curve. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Motion of the Particle
The particle is moving along a curve, and the force \( F \) is constant in magnitude and always acts tangentially to the curve. This means that the force is always in the direction of the particle's instantaneous velocity.

**Hint:** Visualize the particle's path and the direction of the force acting on it.

### Step 2: Work Done by the Force
The work done \( W \) by the force \( F \) as the particle moves along the curve can be expressed as:
\[ W = \int_0^S \mathbf{F} \cdot d\mathbf{s} \]
Since \( \mathbf{F} \) is always tangent to the curve, the angle \( \theta \) between \( \mathbf{F} \) and \( d\mathbf{s} \) is \( 0^\circ \). Therefore, \( \cos(0) = 1 \).

**Hint:** Remember that the work done is calculated using the dot product of force and displacement.

### Step 3: Simplifying the Work Done
Given that \( F \) is constant, we can take it out of the integral:
\[ W = F \int_0^S ds = F \cdot S \]
This shows that the work done depends on the distance \( S \) traveled along the curve.

**Hint:** Recognize that the work done is proportional to the distance traveled when the force is constant.

### Step 4: Path Dependency of Work Done
Since the work done \( W \) depends on the path taken (the distance \( S \)), it indicates that the force does not have a unique potential energy associated with it. In conservative forces, the work done is path-independent, but here it is path-dependent.

**Hint:** Consider the implications of path dependency on the nature of the force.

### Step 5: Conclusion about the Nature of the Force
Since the work done by the force depends on the path taken by the particle, we conclude that the force \( F \) is a non-conservative force.

**Hint:** Recall the definitions of conservative and non-conservative forces to solidify your understanding.

### Final Answer
The force \( F \) must be a non-conservative force.