The figure shows two solid discs with radius $$R$$ and $$r$$ respectively. If mass per unit area is the same for both, what is the ratio of MI of bigger disc around axis $$AB$$ (Which is $$\perp$$ to the plane of the disc and passing through its centre) of MI of smaller disc around one of its diameters lying on its plane? Given $$M$$ is the mass of the larger disc.

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We need to find the ratio of the moment of inertia (MI) of a larger solid disc to that of a smaller solid disc based on the conditions given.

1. Relate Mass to Area

From the problem details, both solid discs have the same mass per unit area (surface mass density, $$\sigma$$):

$$\sigma = \frac{\text{Mass}}{\text{Area}}$$

• Larger Disc (Radius $$R$$):
  Its mass $$M$$ is given by:

  $$M = \sigma \cdot (\pi R^2)$$

• Smaller Disc (Radius $$r$$):
  Its mass $$m$$ is given by:

  $$m = \sigma \cdot (\pi r^2)$$

Taking the ratio of their masses eliminates the constant density $$\sigma$$:

$$\frac{M}{m} = \frac{R^2}{r^2} \implies m = M\frac{r^2}{R^2}$$

2. Formulate the Moments of Inertia

• Larger Disc ($I_1$):
  The moment of inertia of a solid disc around axis $$AB$$ (perpendicular to its plane and passing through its center) is:

  $$I_1 = \frac{1}{2}MR^2$$

• Smaller Disc ($I_2$):
  The moment of inertia of a solid disc around one of its diameters lying on its plane is:

  $$I_2 = \frac{1}{4}mr^2$$

3. Calculate the Ratio ($I_1 : I_2$)

Substitute the expression for the smaller mass $$m$$ into the equation for $$I_2$$:

$$I_2 = \frac{1}{4} \left( M\frac{r^2}{R^2} \right) r^2 = \frac{1}{4}M\frac{r^4}{R^2}$$

Now, write out the explicit ratio of $$I_1$$ to $$I_2$$:

$$\frac{I_1}{I_2} = \frac{\frac{1}{2}MR^2}{\frac{1}{4}M\frac{r^4}{R^2}}$$

Cancel out the mass variable $$M$$ and simplify the fractions:

$$\frac{I_1}{I_2} = \frac{4}{2} \cdot \frac{R^2}{\left(\frac{r^4}{R^2}\right)} = 2 \cdot \frac{R^4}{r^4}$$

Expressing this back as a clean ratio format:

$$I_1 : I_2 = 2R^4 : r^4$$

Conclusion

The ratio of the moments of inertia between the two discs is $$2R^4 : r^4$$, which corresponds exactly to the D option

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