Question

The kinetic energy of the body after the collision is

• A. $\frac{11m{v}^2}{54}$
• B. $\frac{m{v}^2}{108}$
• C. $\frac{17m{v}^2}{54}$
• D. $\frac{17m{v}^2}{108}$

Answer 1

The correct option is A $\frac{11m{v}^2}{54}$

Consider the motion in the COM frame.Also axis of rotation as COM.
$v_{COM}=v/3$
In COM frame: $u_m=2v/3;u_{2}m=-v/3$
When stuck together,COM is at a distance of r on the line joining centres.
angular momentum is conserved in this frame.
$L_i=m\ast 4r/3\ast 2v/3+2m\ast 2r/3\ast v/3$
$L_f=I\omega$
$I_{com}=\left(2/3\right)\left(2m\right)(2r{)}^2+(2m)r^2+(2/3)m{r}^2+m(2r{)}^2$
$I_{com}=12m{r}^2$
$\omega =v/9r$
Total Kinetic Energy = $\left(1/2\right)\left(3m\right)v_{com}^2+\left(1/2\right)I{\omega }^2$
$=3m{v}^2/18+\left(1/2\right)\left(12m{r}^2\right)(v/9r{)}^2$
$=13m{v}^2/54$