HIT THE BALL & THE ROD!!

Classical Mechanics Level 5

A rod of mass $m$ $\&$ length $l$ is hinged at one end $O$ .A particle of mass $m$ travelling with speed $v$ collides with the rod at a distance $x$ from the centre of mass of the rod such that the reaction force at the hinge is $0$ .Then if $x$ = $\frac{l}{\alpha}$ ,evaluate $\alpha$ .

Five identical balls each of mass $m$ $\&$ radius $r$ are stung like beads at random and at rest along a smooth,rigid horizontal thin rod of length $L$ ,mounted between immovable supports.Assume $10r<L$ and that the collison between balls or between balls and the support are elastic.If one ball is stuck horizontally so as to acquire a speed $v$ ,the magnitude of the average force felt by the support is $\frac{\betamv^2}{L-\sigma*r}$ Evaluate $\beta$ and $\sigma$ .

Find $\alpha+\beta+\sigma$

The answer is 17.

1 solution

Rajdeep Brahma
Jun 26, 2018

Relevant wiki: Conservation of Momentum

So in problem 1 linear momentum and angular momentum about hinge both are conserved.Let after collison the speed of the ball be $u_1$ and that of rod be $u_2$ .Conserving momentum we get $v$ = $u_1+u_2$ .Now conserve angular momentum about $O$ .We will get that $v(x+\frac{l}{2})=u_1(x+\frac{l}{2})+u_2*\frac{2l}{3}$ .Which will eventually yield $x=\frac{l}{6}$ .

In problem 2,we know F= $\frac{dP}{dt}$ ,since the collisons are all elastic,when one ball with velocity $v$ hits the other it itself stops and the ball it had hit moves with velocity $v$ .So $dP=2mv$ and $dt=\frac{2(L-10R)}{v})$ and so F= $\frac{mv^2}{L-10*r}$

NOTE: BOTH OF THEM ARE INPHO 2009 PROBLEMS.SO NO CLAIM OF ORIGINALITY IS MADE.