A particle falls from rest from the highest point of a vertical circle of radius r, along a chord without any friction . Show that the time taken by the particle to come down is independent of the chord's length. Find the time in terms of r and g .

Explanation

Let the chord along which the partical falls be CD
[Fig.2.67] . CD makes an angle `theta` with the vertical diameter as shown.
CD =2r `cos theta`. The component of acceleration due to gravity along CD =g`cos theta` . Let the time taken by this particle to fall from rest along CD be t .

Hence, from the equation `s=ut+1/2 at^2`,
`2r cos theta =0+1/2g cos theta*t^2`
`or, t^2 =(4r)/(g) or, t=2sqrt(r/g)`.
The time is independent of `theta`, and hence on the length of the chord CD.