A Problem from my book -2
A mass of is attached to a spring of spring constant . The system can rotate about point . The system is initially at equilibrium and arrangement is vertical. The block is given a velocity minimum so that it just(boundary condition) completes its one revolution around point . Find the eccentricity of path followed by the block about point . ( )
The answer is 0.2.
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1 solution
The solution is very easy. we know that the path will be ellipse as at lowest position spring has max length and at heighest position spring has minimum length and the point O is focus and the length of spring will keep on changing all the time. we have to give minimum velocity to complete one revolution so which means velocity at heighest point must be 0. which means net force towards center must be 0 as no centripital force is required i.e weight balances spring force kx=mg so length of spring = Lo-x ( which is a(1-e).) at lowest position it is in equilibrium i.e kx=mg so length is Lo+x(which is a(1+e). so dividing both equations we get 1-e/1+e = Lo-x/Lo+x. which means e= x/Lo e= mg/kl e=0.2