Question

A solid cylinder of mass $m$ and radius $r$ is rotating about its logitudinal axis (vertical with angular speed $"\omega "$ .If a disc of mass $2m$ and radius $2r$ is gently placed on it coaxially, then the new angular velocity of the system is

• A. $\omega$
• B. $\frac{\omega }{4}$
• C. $\frac{\omega }{8}$
• D. $\frac{\omega }{9}$

Answer 1

The correct option is D $\frac{\omega }{9}$

$\begin{array}{c}\text{hiven-}\ast \text{A solid cylinder mass}=m\text{rodius}=r\\ \text{rotating with angular velocity}\left(w\right)\\ \text{we know}\\ \text{moment of Inertia of solid cylinder}\\ I\text{cylinder}=\frac{m{r}^2}{2}\end{array}$

$\begin{array}{c}\text{We have a disc of mass}=2m\text{radius}=2r\\ \Rightarrow I\text{disc}=\frac{\left(2m\right)\cdot {\left(2r_c\right)}^2}{2}=4{mr}^2\\ \text{we have final moment of Inertia of System -}\end{array}$

$\begin{array}{c}{I}_f=I_{\text{cyuinder}}+I_{\text{disc}}=\frac{m{\mu }^2}{2}+4m{r}^2\\ I_f=\frac{9}{2}m{r}^2\\ \text{By conservation of angular momentum}\end{array}$$\begin{array}{c}\text{By conservation of angular momentum}\\ I_1\cdot w_1=I_f\cdot {\omega }_f\\ \Rightarrow \begin{array}{cc}\frac{m{r}^2}{2}w& =\frac{9m{k}^2}{9},w_f\\ \Rightarrow w_f=\frac{w}{9}& \end{array}\end{array}$