A smooth circular groove has a smooth vertical wall as shown in figure. A block of mass m moves against the wall with a speed v. Which of the following curve represents the correct relation between the normal reaction on the block by the wall (N) and speed of the block (v)?
To find the correct relation between the normal reaction on the block by the wall (\(N\)) and the speed of the block (\(v\)), let's analyze the forces acting on the block in circular motion within the groove:
Concept: When a block moves in a circular path, a centripetal force is required to keep it on its path. This centripetal force is provided by the normal reaction (\(N\)) exerted by the wall. For any block moving in a vertical circular path, we have:
The centripetal force \(F_c\) is given by:
\(F_c = \frac{mv^2}{r}\)
Here, \(m\) is the mass of the block, \(v\) is the speed of the block, and \(r\) is the radius of the circular path.
In this scenario, the normal reaction (\(N\)) by the wall acts as the centripetal force. Therefore:
\(N = \frac{mv^2}{r}\)
This equation shows that the normal force (\(N\)) is directly proportional to the square of the speed (\(v\)). Hence, the relation between \(N\) and \(v\) should be a quadratic relationship.
Conclusion: The curve depicting \(N\) versus \(v\) should be a parabola opening upwards, indicating this quadratic relationship.
From the given options, the graph that correctly represents this relation is:
This graph shows a parabola opening upwards, correctly representing the relationship \(N \propto v^2\).
