Angular momentum L = Iω, thus ω = L/I.

Angular momentum L = Iω = 4 kg·m² * 3 rad/s = 12 kg·m²/s.

Angular momentum L is given by L = Iω. Thus, L = 5 kg·m² * 3 rad/s = 15 kg·m²/s.

Angular momentum L = Iω = 5 kg·m² * 3 rad/s = 15 kg·m²/s.

The rotational kinetic energy is given by KE = 1/2 Iω^2.

The moment of inertia of a solid cylinder about its diameter is I = 1/4 MR^2.

For a solid cylinder, the total kinetic energy is KE_total = KE_translational + KE_rotational = (1/2)mv^2 + (1/2)(1/2)mR^2(ω^2). Since ω = v/R, the translational part is 2/3 of the total.

For a solid disk, the translational kinetic energy is 1/3 of the total kinetic energy when rolling without slipping.

The solid sphere will roll down the incline faster because it has a smaller moment of inertia compared to the hollow sphere.

Both will have the same speed at the bottom due to conservation of energy, as they start from the same height.

The moment of inertia of a solid sphere about an axis through its center is I = 2/5 MR^2.

The moment of inertia of a solid sphere about its center is I = 2/5 MR^2.

Force = Torque / Distance = 15 Nm / 0.3 m = 50 N.

Torque = Force × Distance × sin(θ) => 25 Nm = 10 N × 5 m × sin(θ) => sin(θ) = 0.5 => θ = 30 degrees.

Force = Torque / Radius = 25 Nm / 0.5 m = 50 N.

Force = Torque / Distance = 25 Nm / 0.5 m = 50 N.

Force = Torque / Distance = 30 Nm / 3 m = 10 N.

Angular acceleration α = Torque / Moment of inertia = 5 Nm / 2 kg·m² = 2.5 rad/s².

Force = Torque / Distance = 8 Nm / 0.2 m = 40 N.

The distance traveled by the center of mass after one complete rotation is equal to the circumference of the wheel, which is 2πR.

Torque is maximum when the angle is 90 degrees because τ = F × r × sin(θ) and sin(90°) = 1.

Torque (τ) = F × d × sin(θ) = 15 N × 2 m × sin(90°) = 30 Nm.

If angular momentum is conserved, an increase in moment of inertia results in a decrease in angular velocity, and vice versa.

Angular momentum L = Iω; if L is doubled and I remains constant, ω must also double.

Angular momentum is conserved when the net external torque acting on the system is zero.

Zero angular momentum does not imply rest; it can be in linear motion.

Zero angular momentum implies no net rotation; particles can still move linearly.

Torque is directly proportional to the lever arm; if the lever arm is doubled, the torque also doubles.

Angular momentum L = Iω = 10 kg m² * 5 rad/s = 50 kg m²/s.

Rotational kinetic energy is given by KE = 1/2 I ω² = 1/2 * 10 * 5² = 125 J.