Question

The uniform rod $\mathbf{\left(}\mathbf{\text{length0.60m,mass1.0kg}}\mathbf{\right)}$in Fig. 11-54 rotates in the

plane of the figure about an axis through one end, with a rotational inertia of$\mathbf{0}\mathbf{.12}\mathbf{}{\mathbf{\text{kg.m}}}^{\mathbf{\text{2}}}$

. As the rod swings through its lowest position, it collides with a$\mathbf{\text{0.20kg}}$

putty wad that sticks to the end of the rod. If the rod’s angular speed just before

collision is, $\mathbf{\text{2.4rad/s}}$what is the angular speed of the rod–putty system immediately after collision?

Short answer

Angular speed after collision is$\mathrm{\omega }=1.5\frac{\text{rad}}{\text{s}}.$

Step-by-step solution

Step 1: Given

$\begin{array}{c}\mathrm{m}=1.0\text{kg}\\ {\text{I}}_1=0.12\text{kg}.{\text{m}}^2\\ \mathrm{r}=0.60\text{m}\end{array}$

Determining the concept

First, find the total rotational inertia. Using the law of conservation of angular momentum, find the final angular velocity.According tothe conservation of momentum, momentum of a system is constant if no external forces are acting on the system.

Formula are as follow:

${\mathrm{I}}_{\mathbf{2}}\mathbf{=}{\mathbf{I}}_{\mathbf{1}}\mathbf{+}{\mathbf{mr}}^{\mathbf{2}}$

$\mathbf{Initial}\mathbf{}\mathbf{angular}\mathbf{}\mathbf{momentum}\mathbf{=}\mathbf{Final}\mathbf{}\mathbf{angular}\mathbf{}\mathbf{momentum}$

Where,$\mathit{I}$is moment of inertia,$\mathit{m}$ is mass and $\mathit{r}$is radius.

Determining the angular speed after collision

According to law of conservation of angular momentum:

$\mathrm{Initial}\mathrm{angular}\mathrm{momentum}=\mathrm{Final}\mathrm{angular}\mathrm{momentum}$

$\begin{array}{c}{\mathrm{L}}_1={\mathrm{L}}_2\\ {\mathrm{I}}_1{\mathrm{\omega }}_1={\mathrm{I}}_2{\mathrm{\omega }}_2\end{array}$

$\begin{array}{c}{\mathrm{I}}_2={\mathrm{I}}_1+\mathrm{m}\times {\mathrm{r}}^2\\ {\mathrm{I}}_2=0.12+0.2\times {0.6}^2=0.3192\text{kg}.{\text{m}}^2\\ 0.12\times 2.4=0.192\times {\mathrm{\omega }}_2\end{array}$

By solving above equation for ${\mathrm{\omega }}_2$,

${\mathrm{\omega }}_2=1.5\text{rad}/\text{s}$

Hence,angular speed after collision is$\mathrm{\omega }=1.5\frac{\text{rad}}{\text{s}}.$

Therefore, angular speed after collision can be calculated by using conservation of momentum principle.