A person applies a constant force `vecF` on a particle of mass m and finds tht the particle movs in a circle of radius r with a uniform speed v as seen from an inertial frame of reference.

Options

A. This is not possible
B. There are other forces on the particle
C. The resultant of the other forces is `(mv^2)/r ` towards the centre
D. The resultant of the other forces varies in magnitude as well as in direction.

Answer

To solve the problem, we need to analyze the situation where a constant force \( \vec{F} \) is applied to a particle of mass \( m \), causing it to move in a circular path of radius \( r \) with a uniform speed \( v \). We will evaluate the options provided based on the principles of circular motion and forces acting on the particle.

### Step-by-Step Solution:

1. **Understanding Circular Motion**:
- For an object to move in a circle with a constant speed, there must be a net centripetal force acting towards the center of the circle. This force is given by the formula:
\[ F_c = \frac{mv^2}{r} \]
- Here, \( F_c \) is the centripetal force required to keep the particle moving in a circle of radius \( r \) at speed \( v \).

2. **Identifying Forces Acting on the Particle**:
- The particle experiences several forces:
- The applied force \( \vec{F} \).
- The gravitational force \( \vec{W} = mg \) acting downwards.
- The normal force \( \vec{N} \) acting perpendicular to the surface (if applicable).
- Since the particle is moving in a circle, the net force acting towards the center must equal the centripetal force.

3. **Analyzing the Options**:
- **Option 1**: "This is not possible."
- This is incorrect because it is indeed possible for a particle to move in a circular path under the influence of a constant force.

- **Option 2**: "There are other forces on the particle."
- This is correct. Besides the applied force \( \vec{F} \), there are gravitational and possibly normal forces acting on the particle.

- **Option 3**: "The resultant of the other forces is \( \frac{mv^2}{r} \) towards the center."
- This is incorrect. The net force towards the center must account for all forces acting on the particle, including \( \vec{F} \) and \( mg \). Therefore, we cannot simply state that the resultant force is \( \frac{mv^2}{r} \) without considering the contributions of other forces.

- **Option 4**: "The resultant of the other forces varies in magnitude as well as in direction."
- This is correct. As the particle moves in a circular path, the direction of the applied force \( \vec{F} \) and the gravitational force \( mg \) changes, which affects the resultant force acting on the particle.

### Conclusion:
Based on the analysis, the correct options are:
- Option 2: There are other forces on the particle.
- Option 4: The resultant of the other forces varies in magnitude as well as in direction.