System of Particles and Rotational Motion

Question 1

The Sun rotates around its centre once in 27 days. What will be the period of revolution if the Sun were to expand to twice its present radius without any external influence? Assume the Sun to be a sphere of uniform density.

Accepted Answer

108 days. With no external torque the angular momentum Iω is conserved. For a uniform sphere I = (2/5)MR², so doubling the radius makes I four times larger and therefore ω four times smaller, giving a period of 4 x 27 = 108 days.

Answer

100 days

Answer

105 days

Answer

115 days

Question 2

A uniform rod of mass 20 kg and length 5 m leans against a smooth vertical wall making an angle of 60° with it. The other end rests on a rough horizontal floor. The friction force that the floor exerts on the rod is (Take g = 10 m/s²)

Accepted Answer

100√3 N. A smooth wall can only push horizontally, so friction at the floor is the only force that can balance it. Taking torques about the foot of the rod gives N_wall = (Mg/2) cot θ, where θ = 30° is the angle with the horizontal, so f = N_wall = (200/2)√3 = 100√3 N.

Answer

100 N

Answer

200 N

Answer

200√3 N

Question 3

A solid sphere A of radius R and mass M is attached at a point to a smaller solid sphere B of radius r < R and mass m < M. Assume that the line joining their centres lies along the horizontal. The moment of inertia of the system calculated about a vertical axis passing through the centre of A is IA and that calculated about a vertical axis passing through the centre of B is IB. The difference IA − IB is:

Accepted Answer

(m − M)(R + r)². The centres are a distance (R + r) apart. About A's centre only sphere B is displaced, so the parallel-axis term is m(R + r)²; about B's centre only sphere A is displaced, giving M(R + r)². The two self terms (2/5)MR² and (2/5)mr² appear in both and cancel in the difference, leaving IA − IB = (m − M)(R + r)², which is negative since m < M.

Answer

(M − m)(R + r)²

Answer

(m − M)(R − r)²

Answer

0

Q1.The Sun rotates around its centre once in 27 days. What will be the period of revolution if the Sun were to expand to twice its present radius without any external influence? Assume the Sun to be a sphere of uniform density.

Q2.A uniform rod of mass 20 kg and length 5 m leans against a smooth vertical wall making an angle of 60° with it. The other end rests on a rough horizontal floor. The friction force that the floor exerts on the rod is (Take g = 10 m/s²)