Question

A uniform bar of length 6 a and mass 8 m lies on a smooth horizontal table. Two point masses m and 2m moving in the same horizontal plane with speed 2vand v respectively, strike the bar [as shown in the figure] and stick to the bar after collision. Denoting angular velocity (about the centre of mass), total energy and centre of mass velocity by $?$, E and $v_c$ respectively, we have after collision

• A. $v_c=0$
• B. $?=\frac{3v}{5a}$
• C. $?=\frac{v}{5a}$
• D. $E=\frac{3}{5}mv^2$

Answer 1

Answer: (D)

$E=\frac{3}{5}mv^2$

Answer 2

$P_i=0$
$\therefore P_f=0$ or $v_c=0$
$L_i=L_f or (2mv)a+(2a)(2a)=I?$ ....(i)
Here, $I=\frac{(8m)(6a)^2}{12}+m(2a)^2+(2m)(a)^2=30ma^2$
Substituting in E (i), we get
$?=\frac{v}{5a}$
Further, $E=\frac{1}{2}I?^2=\frac{1}{2}\times(30ma^2)\bigg(\frac{v}{5a}\bigg)^2=\frac{3mv^2}{5}$
$\therefore$ Correct options are (a), (c) and (d).