Question

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the figure is given in the attachement.



A particle is constrained to move on a smooth vertialcircular hoop of radius R. It is projected from the lowestpoint with velocity just sufficient to carry it to the high-est point. Find the time after which the reaction be-tween the particle and the hoop is zero.​

Answer 1

Solution:-The reaction (contact) between the hoop and the ball will be zero at the highest point.....

Now,at highest point of circular hoop

ball will be in vertically downward ,so the gravitational force mg will be in downward direction....

There will be one another force..i.e centripetal...which is eqaul to mv^2\R....that force will be vertically upward,at tthe highest point...

at the highest point,,as normal force is zero,so we can say

mg-mv0^2/R=0

V0=(gR)^1/2

now,

initial veocity=>V0=(gR)^1/2

acceleration=g

final velocity=0

so,

from first equation

v=u+at

0=(gR)^/2 +gt

g^2t^2=gR

t={R/g}^1/2

Hence the time after which reaction between hoop and particle is zero is {R/g}^1/2