Point mass slides from rest from the topmost point of vertical circle of radius r along the smooth chord. Show that the time of descend of mass along the chord is independent of chord chosen. from Physics Motion in A Plane Class 11 Manipur Board
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Point mass slides from rest from the topmost point of vertical circle of radius r along the smooth chord. Show that the time of descend of mass along the chord is independent of chord chosen.

Given, 

Radius of the circle = r 

             

Consider, an arbitrary chord AB which subtends an angle straight theta with vertical.

Now, the length of chord AB is given by, 

AB equals square root of straight r squared plus straight r squared minus 2 rrcos left parenthesis 180 degree minus 2 straight theta right parenthesis end root

   equals square root of 2 straight r squared left parenthesis 1 plus cos space 2 straight theta right parenthesis end root

equals space 2 straight r space cosθ                   

Acceleration of mass along inclined plane is, straight a equals gcosθ

Now, using the equation of motions,

                     straight S equals ut plus 1 half at squared 

∴         2 straight r space cosθ space equals space 0 cross times straight t plus 1 half straight g space cosθt squared 

rightwards double arrow      straight t equals square root of fraction numerator 4 straight r over denominator straight g end fraction end root equals 2 square root of straight r over straight g end root

Since t is independent of straight theta, hence time of descend of mass is independent of the choice of chord. 

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