Final answer:
The moment of inertia of each object can be calculated using specific formulas. For a thin rod, it depends on the axis of rotation. For a solid sphere, it depends on whether it is solid or hollow. For a cylinder, it depends on whether it is thin-walled and hollow or solid.
Step-by-step explanation:
The moment of inertia of a thin rod can be calculated using the formula I = 1/3 * ML^2, where M is the mass of the rod and L is its length. For a rod with a mass 2.50 kg and length of 75.0 cm:
a)(i) When the axis is perpendicular to the rod and passes through one end, the moment of inertia is equal to 1/3 * 2.50 kg * (0.75 m)^2 = 0.46875 kg·m^2.
(ii) When the axis is perpendicular to the rod and passes through the center, the moment of inertia is equal to 1/3 * 2.50 kg * (0.375 m)^2 = 0.234375 kg·m^2.
(iii) When the axis is parallel to the rod and passes through it, the moment of inertia is equal to 1/12 * 2.50 kg * (0.75 m)^2 = 0.15625 kg·m^2.
b)(i) For a solid sphere with a mass of 3.00 kg and radius of 20.0 cm, the moment of inertia is equal to 2/5 * (3.00 kg) * (0.20 m)^2 = 0.96 kg·m^2.
(ii) For a thin-walled hollow shell with the same mass and radius, the moment of inertia is equal to 2/3 * (3.00 kg) * (0.20 m)^2 = 1.28 kg·m^2.
c)(i) For a thin-walled and hollow cylinder with a mass of 8.00 kg and length of 19.5 cm, the moment of inertia can be calculated using the formula I = 1/2 * m * R^2, where R is the radius of the cylinder. Assuming the radius is small, the moment of inertia is approximately equal to 1/2 * 8.00 kg * (0.0975 m)^2 = 0.03705 kg·m^2.
(ii) For a solid cylinder with the same mass and length, the moment of inertia can be calculated using the formula I = 1/2 * m * R^2 + 1/12 * m * L^2, where L is the length of the cylinder. Assuming the radius is small, the moment of inertia is approximately equal to 1/2 * 8.00 kg * (0.0975 m)^2 + 1/12 * 8.00 kg * (0.195 m)^2 = 0.0556125 kg·m^2.