91Ó°ÊÓ

1. The angular speed of the platform,$\text{Ó¬}=1.2\text{rev}/\text{s}$
2. The initial rotational inertia,${\text{I}}_{\text{i}}=6.0\text{kg}.{\text{m}}^2$
3. The new rotational inertia of the system is,${\text{I}}_{\text{f}}=2.0\text{kg}.{\text{m}}^2$

Using the law of conservation of angular momentum, find the new angular speed of the system. Then using this, find the rotational new K.E of the system. Finally, using the rotational K.E. of the system, we can find the ratio of initial and new K.E.

Formula:

$\mathbf{K}\mathbf{.}\mathbf{E}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}{\mathbf{IÓ\neg }}^{\mathbf{2}}$

The law of conservation of angular momentum,${\mathbf{I}}_{\mathbf{i}}{\mathbf{Ó\neg }}_{\mathbf{i}}\mathbf{}\mathbf{=}{\mathbf{I}}_{\mathbf{f}}{\mathbf{Ó\neg }}_{\mathbf{f}}$ .

(a)

According to the law of conservation of angular momentum,

${\text{I}}_{\text{i}}{\text{Ó¬}}_{\text{i}}={\text{I}}_{\text{f}}{\text{Ó¬}}_{\text{f}}$

So, the new angular speed of the system is,

$\begin{array}{c}{\text{Ӭ}}_{\text{f}}=\frac{{\text{I}}_{\text{i}}{\text{Ӭ}}_{\text{i}}}{{\text{I}}_{\text{f}}}\\ ⇒{\text{Ӭ}}_{\text{f}}=\frac{6×1.2}{2}\\ ⇒{\text{Ӭ}}_{\text{f}}=3.6\mathrm{rev}/\mathrm{s}\end{array}$

Therefore, the angular speed of the system after brick is moved closer to the body of the man is ${\text{Ó¬}}_{\text{f}}=3.6\mathrm{rev}/\mathrm{s}$.

(b)

The ratio of the new kinetic energy to the initial $K.E.$of the system is,

$\begin{array}{c}\frac{\text{K}.{\text{E}}_{\text{f}}}{\text{K}.{\text{E}}_{\text{i}}}=\frac{\frac{1}{2}{\text{IӬ}}_{\text{f}}^2}{\frac{1}{2}{\text{IӬi}}_{\text{i}}^2}\\ ⇒\frac{\text{K}.{\text{E}}_{\text{f}}}{\text{K}.{\text{E}}_{\text{i}}}=\frac{\frac{1}{2}\left(2\right){\left(3.6\right)}^2}{\frac{1}{2}\left(6\right){\left(1.2\right)}^2}\\ ⇒\frac{\text{K}.{\text{E}}_{\text{f}}}{\text{K}.{\text{E}}_{\text{i}}}=3\end{array}$

(c)

After pulling the brick closer to the body, the man decreases the rotational inertia of the system which results in the increase in the angular speed. This work done by the man for pulling the brick closer has provided the additional$\mathrm{K}.\mathrm{E}.$