Kinetic energy of rotation is given by KE = (1/2)Iω². If ω is doubled, KE becomes (1/2)I(2ω)² = 4(1/2)Iω² = 4KE = 400 J.

Rotational kinetic energy K = (1/2)Iω² = (1/2)(3 kg·m²)(10 rad/s)² = 150 J.

Angular momentum L = Iω. If I is doubled and ω remains constant, L also doubles.

Using L = Iω, we find ω = L/I = 10/2 = 5 rad/s.

Using L = Iω, we have ω = L/I = 10 kg·m²/s / 2 kg·m² = 5 rad/s.

Angular momentum L = Iω, thus ω = L/I = 12 kg·m²/s / 4 kg·m² = 3 rad/s.

Angular momentum L = Iω, thus ω = L/I = 15 kg·m²/s / 3 kg·m² = 5 rad/s.

Angular momentum L = Iω, if ω is doubled, L becomes 2I(2ω) = 4L.

Angular momentum L = Iω; if I is doubled and ω remains constant, L remains the same.

New angular momentum L' = I'ω' = (1/2I)(2ω) = L.

New angular momentum L' = I'ω' = (1/2 I)(2ω) = Iω = L.

Rotational kinetic energy K = (1/2)Iω² = (1/2)(3 kg·m²)(30 rad/s)² = 450 J.

Angular deceleration = (final angular velocity - initial angular velocity) / time = (0 - 30 rad/s) / 5 s = -6 rad/s².

Angular momentum is conserved in the absence of external torque.

Using L = Iω, we have ω = L/I = 10/2 = 5 rad/s.

Angular momentum L = Iω, if ω is doubled, L becomes 2I(2ω) = 4L.

Angular momentum L = Iω; if ω is doubled, L becomes 2I(2ω) = 4L.

Angular momentum L is proportional to angular velocity, so if angular velocity is doubled, angular momentum becomes 4L.

Angular momentum L = mvr, where v is the orbital speed and r is the radius of the orbit.

Using conservation of energy, the potential energy mgh converts to kinetic energy. For a solid cone, the final speed at the bottom is v = √(2gh).

For a solid cone rolling down an incline, the potential energy at height h is converted into translational and rotational kinetic energy, leading to PE = 2KE.

The moment of inertia of a solid cone about its axis is I = (1/10)mR^2.

The solid cylinder has a smaller moment of inertia compared to the hollow cylinder, thus it will have a greater speed at the bottom.

The solid cylinder reaches the bottom first because it has a smaller moment of inertia compared to the hollow cylinder.

Using conservation of energy, potential energy at the top equals kinetic energy at the bottom. The speed v at the bottom is v = √(2gh).

At the bottom, total kinetic energy = translational + rotational. For a solid cylinder, the ratio of translational to total kinetic energy is 2:1.

For a solid cylinder, the ratio of translational kinetic energy to total kinetic energy is 2/5.

At the bottom, total mechanical energy is converted into kinetic energy, which is the sum of translational and rotational kinetic energy. For a solid cylinder, 2/3 of the energy is kinetic.

Using conservation of energy, potential energy at the top equals kinetic energy at the bottom. For a solid cylinder, v = √(3gh/2).

The solid sphere will have a greater translational speed because it has a smaller moment of inertia.