Question

A horizontal vinyl record of mass $\mathbf{0}\mathbf{.10}\mathbf{}\mathbf{kg}$ and radius$\mathbf{0.10}\mathbf{}\mathbf{m}$rotates freely about a vertical axis through its centre with an angular speed of$\mathbf{4.7}\mathbf{}\mathbf{rad}\mathbf{/}\mathbf{s}\mathbf{.}$The rotational inertia of the record about its axis of rotation is$\mathbf{5}\mathbf{.0}\mathbf{}\mathbf{\times }\mathbf{}{\mathbf{10}}^{\mathbf{-}\mathbf{4}\mathbf{}}\mathbf{kg}\mathbf{}\mathbf{\cdot }{\mathbf{m}}^{\mathbf{2}}\mathbf{.}$A wad of wet putty of mass$\mathbf{0}\mathbf{.020}\mathbf{}\mathbf{kg}\mathbf{.}$drops vertically onto the record from above and sticks to the edge of the record. What is the angular speed of the record immediately after the putty sticks to it?

Short answer

Angular speed of record immediately after dropping the putty on the record is$\mathrm{\omega }=3.\text{4rad/s.}$

Step-by-step solution

Step 1: Given

$\begin{array}{c}{\mathrm{\omega }}_{\mathrm{i}}=4.7\text{rad}/\text{s}\\ {\mathrm{I}}_{\mathrm{i}}=5\times {10}^{-4}{\text{kg.m}}^{\text{2}}\\ \mathrm{M}=0.0\text{2kg}\\ \mathrm{R}=0.10\text{m}\end{array}$

Determining the concept

Find the initial angular momentum of the record before dropping the putty on it. As angular momentum is conserved when no external torque acts on it, find the final angular velocity of the record after dropping the putty on it.According tothe conservation of momentum, momentum of a system is constant if no external forces are acting on the system.

Formula is as follow:

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

Determining the angular speed of record immediately after dropping the putty on the record

According to law of conservation of momentum:

$\text{Initialangularmomentum}=\text{Finalangularmomentum}$

$\begin{array}{c}{\mathrm{L}}_{\mathrm{i}}={\mathrm{L}}_{\mathrm{f}}\\ {\mathrm{I}}_{\mathrm{i}}{\mathrm{\omega }}_{\mathrm{i}}={\mathrm{I}}_{\mathrm{f}}{\mathrm{\omega }}_{\mathrm{f}}\\ {\mathrm{I}}_{\mathrm{f}}={\mathrm{I}}_{\mathrm{i}}+{\mathrm{mR}}^2=5\times {10}^{-4}+(0.02\times {0.1}^2)=7\times {10}^{-4}{\text{kg.m}}^{\text{2}}\\ 5\times {10}^{-4}\times 4.7=(7\times {10}^{-4})\times {\mathrm{\omega }}_{\mathrm{f}}\end{array}$

${\mathrm{\omega }}_{\mathrm{f}}=3.4\text{rad/s}$

Hence,the angular speed of record immediately after dropping the putty on the record is$\mathrm{\omega }=3.\text{4rad/s.}$

Therefore, by applying law of conservation of angular momentum, the final angular speed can be calculated.