A block of mass m slides down a smooth vertical circular track. During the motion, the block is in

Options

A. vertical equilibrium
B. horizontl equilibrium
C. radia equilibrium
D. none of these

Answer

To solve the question about the block of mass \( m \) sliding down a smooth vertical circular track, we need to analyze the motion of the block and the forces acting on it.

### Step-by-Step Solution:

1. **Understanding the Motion**:
- The block is sliding down a vertical circular track. This means it is moving along a circular path under the influence of gravitational force.

2. **Identifying Equilibrium Conditions**:
- For an object to be in equilibrium, the net force acting on it must be zero. This applies to vertical, horizontal, and radial (centripetal) directions.

3. **Vertical Equilibrium**:
- In vertical equilibrium, the acceleration in the vertical direction must be zero. However, as the block slides down, it is influenced by gravitational force \( mg \) acting downwards, and there will be a normal force acting perpendicular to the surface of the track. Therefore, the net force in the vertical direction is not zero.

4. **Horizontal Equilibrium**:
- In horizontal equilibrium, the acceleration in the horizontal direction must also be zero. However, as the block moves along the circular path, it experiences a change in direction, which means there is a centripetal acceleration directed towards the center of the circle. This indicates that the net force in the horizontal direction is not zero.

5. **Radial Equilibrium**:
- Radial equilibrium refers to the balance of forces in the radial direction (towards the center of the circle). The block experiences a centripetal acceleration given by \( a_r = \frac{v^2}{r} \), where \( v \) is the velocity of the block and \( r \) is the radius of the circular path. Since the block is moving, \( v \) is not zero, and thus the radial acceleration cannot be zero.

6. **Conclusion**:
- Since the block is not at rest and is continuously accelerating due to gravity and the circular motion, it is clear that the block is not in vertical equilibrium, horizontal equilibrium, or radial equilibrium. Therefore, the correct answer is "none of these."

### Final Answer:
The block is in none of these equilibria.