Moment of inertia depends on distribution of mass around axis...the more the mass near the axis less the moment of inertia...four rods forming a square has less more moment of inertia because of less mass ner axis

Moment of Inertia:

The moment of inertia of a body is a measure of its resistance to rotational motion about a particular axis. It depends on the distribution of mass within the body and the axis of rotation. The moment of inertia is given by the formula:

I = ∫ r^2 dm

where I is the moment of inertia, r is the perpendicular distance from the axis of rotation to the element of mass dm, and the integral is taken over the entire mass of the body.

Comparison of Options:

To determine which option will have the largest moment of inertia, we need to compare the distributions of mass and the distances of the mass elements from the axis of rotation for each option.

a) Disc of radius a:
- The mass of the disc is distributed uniformly throughout its volume.
- The distance of the mass elements from the axis of rotation is given by the radius of the disc, which is a.
- The moment of inertia of the disc can be calculated using the formula for a solid disc: I = (1/2) M a^2, where M is the total mass of the disc.

b) Ring of radius a:
- The mass of the ring is also distributed uniformly throughout its volume.
- The distance of the mass elements from the axis of rotation is given by the radius of the ring, which is a.
- The moment of inertia of the ring can be calculated using the formula for a hollow cylinder: I = M a^2, where M is the total mass of the ring.

c) Square lamina of side 2a:
- The mass of the square lamina is concentrated at its vertices.
- The distance of the mass elements from the axis of rotation is given by the distance from the center of mass of the square to its vertices, which is a.
- The moment of inertia of the square lamina can be calculated using the parallel axis theorem, which states that the moment of inertia about an axis parallel to and a distance d from an axis through the center of mass is given by I' = I + Md^2, where I is the moment of inertia about the axis through the center of mass, M is the total mass, and d is the distance between the axes.
- In this case, the moment of inertia about the axis through the center of mass is zero, as the mass is concentrated at the vertices. Therefore, the moment of inertia of the square lamina is given by I' = Ma^2.

d) Four rods forming a square of side 2a:
- The mass of each rod is distributed uniformly along its length.
- The distance of the mass elements from the axis of rotation is given by the distance from the center of mass of each rod to the axis of rotation, which is a/√2.
- The moment of inertia of each rod can be calculated using the formula for a thin rod rotating about an axis perpendicular to its length: I = (1/3) ML^2, where M is the mass of the rod and L is its length.
- The moment of inertia of the four rods forming a square can be calculated by summing the moments of inertia of each rod. Since the rods are identical, the total moment of inertia is given by I = 4(1/3) Ma^2/2 = (2/3) Ma^2.

Conclusion:

Comparing the moment of inertia values for each option, we find:
- Moment