Question

A raw egg and hard boiled eggs are made to spin on a table with the same angular momentum about the same axis. The ratio of the time taken by the two to stop is:
A. \[=1\]
B. \[<1\]
C. \[>1\]
D. None of these

Answer 1

Hint: The moment of inertia depends on the distribution of the mass about the axis of rotation of the body. When torque is applied to the body then it produces the angular acceleration in the body. If the torque applied is in the opposite direction of the angular velocity then the body slows down. If the torque applied is in the same direction of the angular velocity then the body speeds up.

Complete step by step solution:
The moment of inertia of a rigid body is given as the sum of the product of mass with the square of the distance from the axis of the rotation.
$MI=\sum\limits_{i=1}^{i=n}{{{m}_{i}}r_{i}^{2}}$
A raw egg is like a spherical shell & a hard-boiled egg is like a solid sphere.
The moment of inertia of the spherical shell is ${{I}_{1}}$$=$$\dfrac{2}{3}M{{R}^{2}}$
And the moment of inertia of the solid sphere is ${{I}_{2}}$$=$$\dfrac{2}{5}M{{R}^{2}}$
Angular momentum of the body is given as,
$L=I\omega $
Where,
$I=$Moment of inertia of the body
$\omega =$Angular speed of the body
It is given that the angular momentum of raw egg is equal to the hard-boiled egg.
$\begin{align}
  & {{L}_{1}}={{L}_{2}} \\
 & {{I}_{1}}{{\omega }_{1}}={{I}_{2}}{{\omega }_{2}} \\
\end{align}$
As ${{I}_{1}}>{{I}_{2}}$,
Hence, ${{\omega }_{2}}>{{\omega }_{1}}\ldots \ldots \left( i \right)$
From the first equation of angular motion,
\[{{\omega }_{f}}={{\omega }_{0}}+\alpha t\]
Where,
$\alpha $$=$The acceleration for both cases
${{\omega }_{o}}=$Initial angular speed of the body
${{\omega }_{f}}=$Final angular speed of the body$=0$ for both the cases
then,
$\dfrac{{{t}_{1}}}{{{t}_{2}}}=\dfrac{\left( \dfrac{{{\omega }_{1}}}{\alpha } \right)}{\left( \dfrac{{{\omega }_{2}}}{\alpha } \right)}=\dfrac{{{\omega }_{1}}}{{{\omega }_{2}}}\ldots \ldots \left( ii \right)$
From equation $\left( i \right)$ and equation$\left( ii \right)$
$\dfrac{{{t}_{1}}}{{{t}_{2}}}<1$

Thus option B is correct.

Note: - The moment of inertia of the solid sphere is less than the moment of inertia of the spherical shell.
- The angular acceleration is directly proportional to the torque applied on the body.