Question

Of the two eggs which have identical sizes, shapes and weights, one is raw and other is boiled. The ratio between the moment of inertia of the raw to the boiled egg about the central axis is –

& \text{A) }=1 \\\ & \text{B) > 1} \\\ & \text{C) < 1} \\\ & \text{D) Not comparable} \\\ \end{aligned}$$

Answer 1

We need to understand the different parameters involved in the moment of inertia which can differ for a boiled egg and a raw egg. The ratio between them can vary only if any of the physical quantities change after boiling the same egg.

Complete step by step solution:
The moment of inertia is the rotational analogue of the mass of an object. An object in rest or undergoing a rectilinear motion considers its mass along the different directions with respect to the horizontal, whereas an object undergoing a circular or rotational motion with respect to any of its axes have a tendency to rotate with velocities depending on the distance of the mass from the center of rotation.
The moment of inertia for a system of mass M whose center of mass is at a distance R from its axis is given generally as –
$I=M{{R}^{2}}$
The moment of inertia is highly dependent on the position of center of mass of the system. The moment of inertia will change for the same system if the axis of rotation considered is changed or the position of the center of mass is changed.
Now, let us consider the situations given to us. The first one is where the moment of inertia of a raw egg is considered. Let us assume that the egg is symmetrical and spherical in shape. The axis of rotation will be the same along any axis as the distance from the center of mass is uniform.

Also, for a raw egg the center of mass is near the middle of the egg as the mass is distributed uniformly in the egg. Let ‘r’ be this position from the axis. The center of mass of raw egg is denoted by red lines.
Now, the boiled eggs have a mass concentrated at its center compared to raw egg as seen in the figure marked by yellow line of radius r’.
We understand that the ratio between the moment of inertia of the raw egg and the boiled egg will be a value greater than 1 as –

& I\propto {{R}^{2}} \\\ & \Rightarrow \dfrac{I}{I'}=\dfrac{{{r}^{2}}}{r{{'}^{2}}} \\\ & \text{but,} \\\ & r>r' \\\ & \Rightarrow \dfrac{r}{r'}>1 \\\ & \text{and,} \\\ & \therefore \dfrac{I}{I'}>1 \\\ \end{aligned}$$ Therefore, we can conclude that the ratio will be greater than 1. **So, the correct answer is “Option B”.** **Note:** The shape of the egg in this case is not a problem as the raw egg and boiled egg have the same shape. The moment of inertia along any axis passing through the center of these eggs will give the ratio to be always greater than one as we got in this.