91Ó°ÊÓ

1. Mass of the bigger disk is$10\text{m}$
2. Radius of the larger disk is$3\mathrm{r}$
3. Mass of the smaller disk is$\mathrm{m}$
4. Radius of the smaller disk is$\mathrm{r}$
5. ${\mathrm{I}}_{\mathrm{disk}}=30{\text{0kg.m}}^{\text{2}}$

Calculate the initial and final moment of inertia by using the formula for the rotational inertia of the disk. Use law of conservation of momentum to find the final angular speed. Then, calculate the ratio of final to initial kinetic energies.According tothe conservation of momentum, momentum of a system is constant if no external forces are acting on the system.

Formula are as follow:

$\begin{array}{l}\mathbf{I}\mathbf{=}\frac{{\mathbf{MR}}^{\mathbf{2}}}{\mathbf{2}}\\ \mathbf{Initial}\mathbf{}\mathbf{angular}\mathbf{}\mathbf{momentum}\mathbf{=}\mathbf{Final}\mathbf{}\mathbf{angular}\mathbf{}\mathbf{momentum}\end{array}$

Where,Iis moment of inertia, M is mass and R is radius.

(a)

Using the formula for the rotational inertia of the disk, find the initial rotational inertia of the system,

$\begin{array}{l}{\mathrm{I}}_{\mathrm{i}}={\mathrm{I}}_{\mathrm{bigdisk}}+{\mathrm{I}}_{\mathrm{smalldisk}}\\ \mathrm{I}=\frac{{\mathrm{MR}}^2}{2}+\frac{{\mathrm{mr}}^2}{2}=\frac{10\mathrm{m}×9{\mathrm{r}}^2}{2}+\frac{{\mathrm{mr}}^2}{2}=\frac{91{\mathrm{mr}}^2}{2}\end{array}$

The new rotational inertia of system is,

${\text{I}}_{\text{f}}=\frac{{\mathrm{MR}}^2}{2}+\frac{{\mathrm{mr}}^2}{2}+\mathrm{m}{(\mathrm{R}−\mathrm{r})}^2=\frac{99}{2}{\mathrm{mr}}^2$

By law of conservation of angular momentum,

$\begin{array}{c}{\mathrm{L}}_{\mathrm{i}}={\mathrm{L}}_{\mathrm{f}}\\ {\mathrm{I}}_{\mathrm{i}}{\mathrm{Ó\neg }}_{\mathrm{i}}={\mathrm{I}}_{\mathrm{f}}{\mathrm{Ó\neg }}_{\mathrm{f}}\\ \frac{91{\mathrm{mr}}^2}{2}×20=\frac{99}{2}{\mathrm{mr}}^2×{\mathrm{Ó\neg }}_{\mathrm{f}}\\ {\mathrm{Ó\neg }}_{\mathrm{f}}=18\text{rad/s}\end{array}$

Hence, angular velocity about the center of the larger disk is, $\mathrm{Ó\neg }=18\frac{\text{rad}}{\text{s}}.$

(b)

Now,

$\frac{{\mathrm{I}}_{\mathrm{f}}}{{\mathrm{I}}_{\mathrm{i}}}=\frac{99}{91}$

And,

$\frac{{\mathrm{Ó\neg }}_{\mathrm{f}}}{{\mathrm{Ó\neg }}_{\mathrm{i}}}=\frac{91}{99}$

Thus,

$\frac{{\mathrm{KE}}_{\mathrm{f}}}{{\mathrm{KE}}_{\mathrm{i}}}=\frac{{\mathrm{I}}_{\mathrm{f}}{\mathrm{Ó\neg }}_{\mathrm{f}}^2}{{\mathrm{I}}_{\mathrm{i}}{\mathrm{Ó\neg }}_{\mathrm{i}}^2}=\left(\frac{99}{91}\right)×{\left(\frac{91}{99}\right)}^2=0.92$

Hence, the ratio of new kinetic energy to initial kinetic energy is $\mathrm{0.92.}$

Therefore, using the formula for rotational inertia and conservation of angular momentum, the ratio of kinetic energies can be found.