The linear velocity v of the center of the wheel is related to the angular velocity by v = Rω.

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

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

The total moment of inertia of a composite body about the same axis is the sum of the individual moments of inertia.

The moment of inertia is smallest when all mass is concentrated at the center, as it minimizes the distance from the axis of rotation.

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

The moment of inertia of a rectangular plate about an axis through its center parallel to side a is I = 1/12 Mb^2.

The moment of inertia of a solid disk about an axis through its center is I = 1/2 MR^2.

The moment of inertia for a system of particles is calculated as I = Σ(m_i * r_i^2), where m_i is the mass and r_i is the distance from the axis.

This is known as the Parallel Axis Theorem, which states that I = Σ(m_i * r_i^2).

The moment of inertia is calculated by summing the products of mass and the square of the distance from the axis of rotation.

The total moment of inertia for a system of particles is the sum of each particle's moment of inertia, I_total = Σ(m_i * r_i^2).

The total moment of inertia for a system of particles is calculated by adding the individual moments of inertia.

The moment of inertia of a thin circular ring about an axis through its center and perpendicular to its plane is I = MR^2.

Angular acceleration α = τ/I = 5 N m / 15 kg m² = 0.33 rad/s².

Angular momentum L = Iω; if I is halved and ω remains constant, L remains L.

As the child moves closer, the moment of inertia decreases, and to conserve angular momentum, angular velocity must increase.

Angular momentum remains constant if no external torque acts, according to the conservation of angular momentum.

As the child moves towards the center, the moment of inertia decreases, thus angular velocity increases to conserve angular momentum.

For rolling without slipping, the linear velocity v is equal to the radius r times the angular velocity ω, hence v = rω.

Torque = Force × Distance × sin(θ) = 12 N × 1 m × sin(30°) = 12 N × 1 m × 0.5 = 6 Nm.

Torque (τ) = F × r × sin(θ) = 15 N × 1.5 m × sin(30°) = 15 × 1.5 × 0.5 = 11.25 Nm.

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

The solid cylinder has a lower moment of inertia compared to the hollow cylinder, thus it accelerates faster down the incline.

The solid sphere has a smaller moment of inertia compared to the hollow sphere, thus it accelerates faster and reaches the bottom first.

The angular momentum of a rotating planet is constant if no external torques act on it.

The total kinetic energy of a rolling object is the sum of translational and rotational kinetic energy, which can be expressed as (1/2)mv^2 + (1/2)(2/5)mr^2(ω^2).

The total kinetic energy of a rolling object is the sum of translational and rotational kinetic energy, which simplifies to (1/2)mv^2 + (1/4)mv^2 = (3/4)mv^2.

The linear velocity v of a rolling object is given by v = Rω.

For rolling without slipping, the relationship is v = ωR, where v is the translational speed and ω is the angular speed.