Question

A uniform bar of length ${}^{\prime }6l$ lies on a smooth horizontal table. Two point masses m and 2m moving in the same horizontal plane with speed 2v and v respectively.strike the bar (as shown in the fig.) and stick to the bar after collision. Total center of mass, c) will be :-

Answer 1

$F_{ext}=0$

$-2mv+m\times 2v+0=\left(2m+m+8m\right)v_c$

$v_c=0$

$zext=0$

$m_1{v}_1{r}_1+m_2{v}_2{r}_2=\left(J_1+J_1+J_1\right)w$

$2mva+2m\left(2v\right)\left(2a\right)=$

$2m{a}^2+m{\left(2a\right)}^2+\frac{8m\times {\left(6a\right)}^2}{12}w$

$6mva=30mw{a}^2$

$w=\frac{v}{5a}$

$E=\frac{1}{2}I{w}^2=\frac{1}{2}\left(30m{a}^2\right){\left(\frac{v}{2a}\right)}^2$

$=\frac{3}{5}m{v}^2$