Question

A gramophone record has radius r and mass M. It
is rotating with a speed . Two drops of wax of
mass m fall on it at a distance r/2 on opposite sides.
The new angular speed is​

Answer 1

Question:-

A gramophone record has radius $\displaystyle\sf {R}$ and mass $\displaystyle\sf {M.}$ It is rotating with an angular speed $\displaystyle\sf {\omega.}$ Two drops of wax of mass $\displaystyle\sf {m}$ fall on it at a distance $\displaystyle\sf {\dfrac {R}{2}}$ from the center of the record on opposite sides. Find the new angular speed.

Answer:-

$\displaystyle\large\boxed {\sf {\omega'=\left (\dfrac {M}{M+m}\right)\omega}}$

Solution:-

The moment of inertia of the gramophone record is,

$\displaystyle\longrightarrow\sf{I_1=\dfrac {1}{2}MR^2}$

since it is a disc.

The two wax drops were initially at rest. Since they're small drops, their moment of inertia can be taken as $\displaystyle\sf {\dfrac {1}{4}mR^2}$ each. $\displaystyle\sf {\dfrac {1}{4}R^2=\left (\dfrac {R}{2}\right)^2}$ is because the wax drops are placed at a distance $\displaystyle\sf {\dfrac {R}{2}}$ from the center of the gramophone record on opposite sides.

$\displaystyle\longrightarrow\sf{I_2=2m\left (\dfrac {R}{2}\right)^2}$

$\displaystyle\longrightarrow\sf{I_2=\dfrac {1}{2}mR^2}$

where $\displaystyle\sf {I_2}$ is the total moment of inertia of the two wax drops.

Let the new angular speed of the system be $\displaystyle\sf {\omega',}$ which has to be found out in connection with $\displaystyle\sf {\omega.}$

By law of conservation of angular momentum (no external torque is present),

$\displaystyle\longrightarrow\sf{I_1\omega+ I_2(0)=(I_1+I_2)\omega'}$

$\displaystyle\longrightarrow\sf{\omega'=\dfrac {I_1\omega}{I_1+I_2}}$

$\displaystyle\longrightarrow\sf{\omega'=\dfrac {\dfrac {1}{2}MR^2\omega}{\dfrac {1}{2}MR^2+\dfrac {1}{2}mR^2}}$

$\displaystyle\longrightarrow\sf{\underline {\underline {\omega'=\left (\dfrac {M}{M+m}\right)\omega}}}$

Suppose $\displaystyle\sf{m\ \textless\textless\ M.}$ Then,

$\displaystyle\longrightarrow\sf{\underline{\underline {\omega'\simeq \omega}}}$

This means the initial angular speed of the record will have a small change if the wax drops are dropped on it.