Question

A body crosses the topmost point of a vertical circle with critical speed. What will be its centripetal acceleration when the ring is horizontal?
A) $g$
B) $2g$
C) $3g$
D) $6g$

Answer 1

Hint: Use the formula for critical speed to get the speed at the top (in terms of variables only). This will be the initial speed for when the string becomes horizontal, as mentioned in the question. Use the formula for speed at this horizontal point, and put values and solve to get the final answer.

Formula Used:
Critical speed at the top of a vertical circle, ${v_c} = \sqrt {gl} $ where, $g$ is acceleration due to gravity, and $l$ is radius of the vertical circle.
At a point in horizontal circle, when the string becomes horizontal, the speed of object, ${v^2} = {u^2} + 2gh$ where, $u$ is the initial speed of the object, $g$ is acceleration due to gravity and $h$ is the change in height of the object.

Complete Step by Step Solution:
It is given that the object at the top of the vertical circle has speed equal to critical speed.
Therefore, the speed of the object at the topmost point $ = \sqrt {gl} $ where, $g$ is acceleration due to gravity, and $l$ is radius of the vertical circle. This will act as initial speed for when the string becomes horizontal. i.e. $u = \sqrt {gl} $
Now, when the object reaches at the point where string becomes horizontal, its speed is given by ${v^2} = {u^2} + 2gh$ where, $u$ is the initial speed of the object, $g$ is acceleration due to gravity and $h$ is the change in height of the object.
Therefore, speed of the object at this point, ${v^2} = {(\sqrt {gl} )^2} + 2gh$ (as mentioned above)
${v^2} = gl + 2gl$ (the change in height of the object from initial point is equal to $l$ )
$ \Rightarrow {v^2} = 3gl$
Now, the centripetal acceleration will be ${a_c} = \dfrac{{{v^2}}}{l} = \dfrac{{3gl}}{l} = 3g$

Therefore, option (C) is the final answer.

Note: The critical speed can be defined as the minimum speed required by the mass tied to the string in a vertical circle at the topmost point of the vertical circle to complete the circular motion. The centripetal acceleration is the acceleration of the body towards the center along the radius. It will be different for both vertical and horizontal alignment of the ring because the right direction is different in both the cases.