A small block is shot into each of the four track as shown in Fig 15.7.1. Each of the track rises to the same height. The speed with which the block enters the track is the same in all cases. At the highest point of the track, the normal reaction is maximum in

Options

A.
B.
C.
D.

Explanation

To solve the problem of determining which track has the maximum normal reaction at the highest point, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Forces at the Highest Point**:
At the highest point of the track, the forces acting on the block are:
- The gravitational force (weight) acting downward, \( mg \).
- The normal reaction force \( N \) acting downward.

2. **Centripetal Force Requirement**:
For the block to move in a circular path, there must be a net inward (centripetal) force. This force is provided by the normal force and the weight of the block. The centripetal force required is given by:
\[ F_c = \frac{mv^2}{r} \]
where \( m \) is the mass of the block, \( v \) is the speed of the block, and \( r \) is the radius of curvature of the track at that point.

3. **Setting Up the Equation**:
At the highest point, the centripetal force is provided by the normal force and the weight of the block:
\[ \frac{mv^2}{r} = N + mg \]
Rearranging gives us the expression for the normal force:
\[ N = \frac{mv^2}{r} - mg \]

4. **Maximizing the Normal Force**:
To maximize the normal force \( N \), we need to minimize the radius \( r \) of the track. The smaller the radius, the larger the value of \( \frac{mv^2}{r} \) will be, thus increasing \( N \).

5. **Comparing Different Tracks**:
Since all tracks rise to the same height and the speed of the block entering each track is the same, we need to analyze the radius of curvature for each track. The track with the smallest radius of curvature will yield the maximum normal reaction.

6. **Conclusion**:
Based on the analysis, the track with the smallest radius of curvature will have the maximum normal reaction at the highest point.

### Final Answer:
The maximum normal reaction occurs in the track with the smallest radius of curvature.