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

A particle falls from rest from thehighest point ofa vertical circle o

1.	A particle falls from rest from thehighest point ofa vertical circle of radius r, along a chord without any friction . Show thatthe time taken by the particle to come down is independent of the chord's length. Find the time in terms of r and g .
Answer» Solution :Let the chord ALONG which the partical falls be CD [Fig.2.67] . CDmakes an angle `theta` with the vertical diameteras shown. CD =2r `cos theta`. The component of acceleration DUE togravity 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 onthelengthof the chord CD.