91Ó°ÊÓ

$\begin{array}{l}\mathrm{L}=0.6\text{m}\\ \mathrm{Ó\neg }=80\text{rad}/\text{sec}\\ \mathrm{v}=40\text{m/s}\end{array}$

Use law of conservation of angular momentum to find the distance. Also, use the formula for angular momentum in terms of mass, velocity and distance to find the angular momentum.According tothe conservation of momentum, momentum of a system is constant if no external forces are acting on the system.

Formula are as follow:

$\begin{array}{l}{\mathbf{L}}_{\mathbf{i}}\mathbf{=}{\mathbf{L}}_{\mathbf{f}}\\ \mathbf{L}\mathbf{=}\mathbf{mvr}\end{array}$

Where,$\mathbf{L}$is angular momentum, $\mathbf{m}$is mass, $\mathbf{V}$is velocity and $\mathbf{r}$ is radius.

(a)

From the law of conservation of angular momentum,

$\begin{array}{c}{\mathrm{L}}_{\mathrm{i}}={\mathrm{L}}_{\mathrm{f}}\\ −\mathrm{dmv}+\frac{1}{12}{\mathrm{ML}}^2\mathrm{Ó\neg }=0\end{array}$

Negative sign is taken for the clockwise rotation.

So,

$\mathrm{d}=\frac{{\mathrm{ML}}^2\mathrm{Ó\neg }}{12\mathrm{mv}}$

So,

$\begin{array}{l}\mathrm{d}=\frac{\mathrm{M}×{0.6}^2×80}{12×\frac{\mathrm{M}}{3}×40}\\ \mathrm{d}=0.18\text{0m}\end{array}$

Hence, the distance $\mathrm{d}$is$0.180\text{m.}$

(b)

If the length$\mathrm{d}$ is increased, system rotates clockwise. This is because, when$d$ increases, term$mvd$in above equation would become more negative and hence would cause the clockwise rotation as per the sign conventions.

Hence, increasing the value of $d$would cause the rod to rotate in clockwise direction.

Therefore, using the law of conservation of angular momentum and angular momentum formula, the required distance and the sense of rotation can be found if the direction is changed.