A4 Rigid Body Mechanics

What is moment of inertia?

A measure of an object's resistance to angular acceleration about a given axis.

What is the equation for moment of inertia?

I = Σmr², where m is each mass and r is its perpendicular distance from the axis.

What is the SI unit of moment of inertia?

kg m²

What factors affect moment of inertia?

The mass of an object and how its mass is distributed relative to the axis of rotation.

What is the relationship between linear and rotational displacement?

Displacement s corresponds to angular displacement θ, with s = rθ.

What is the relationship between linear and angular velocity?

v = rω

What is the relationship between linear and angular acceleration?

a = rα

What is the relationship between linear force and torque?

Force F corresponds to torque τ.

What is the relationship between mass and moment of inertia?

Mass m corresponds to moment of inertia I.

What is the relationship between linear momentum and angular momentum?

Linear momentum p = mv corresponds to angular momentum L = Iω.

What is angular acceleration?

The rate of change of angular velocity with time.

What is the equation for angular acceleration?

α = Δω/Δt

What is the SI unit of angular acceleration?

rad s⁻².

What is the equation for angular velocity?

ω = Δθ/Δt

What is the SI unit of angular velocity?

rad s⁻¹

What is the equation for uniformly angularly accelerated motion (1)?

ω = ω₀ + αt

What is the equation for uniformly angularly accelerated motion (2)?

θ = ω₀t + ½αt²

What is the equation for uniformly angularly accelerated motion (3)?

θ = ½(ω₀ + ω)t

What is the equation for uniformly angularly accelerated motion (4)?

ω² = ω₀² + 2αθ

What is rotational kinetic energy?

The kinetic energy possessed by an object due to its rotation.

What is the equation for rotational kinetic energy?

Ek = ½Iω²

What is torque?

The turning effect of a force about an axis or pivot.

What is the equation for torque?

τ = rF sin θ, where r is the distance from the axis and θ is the angle between r and F.

What is the SI unit of torque?

N m

When is torque maximised?

When the force acts perpendicular to the lever arm (θ = 90°).

What is the moment arm?

The perpendicular distance from the axis of rotation to the force's line of action.

What is a couple?

Two equal, opposite and parallel forces acting along different lines, producing rotation without resultant force.

What is the torque of a couple?

τ = Fd, where d is the perpendicular separation between the forces.

What is rotational equilibrium?

When the resultant torque acting on an object is zero.

What are the conditions for complete equilibrium?

Resultant force = 0 and resultant torque = 0.

What is Newton's second law for rotational motion?

Στ = Iα, for a rigid body rotating about a fixed axis with constant moment of inertia.

What happens when there is a resultant torque?

The object experiences angular acceleration.

What is angular momentum?

The rotational equivalent of linear momentum.

What is the equation for angular momentum?

L = Iω, for a rigid body rotating about a fixed axis.

What is the SI unit of angular momentum?

kg m² s⁻¹

What is the principle of conservation of angular momentum?

Angular momentum remains constant when the resultant external torque is zero.

Why does a spinning skater rotate faster when pulling in their arms?

Their moment of inertia decreases, so angular velocity increases to conserve angular momentum.

What is angular impulse?

The product of resultant torque and the time interval over which it acts.

What is the equation for angular impulse?

Angular impulse = τΔt, for constant torque.

What is the relationship between angular impulse and change in angular momentum?

τΔt = ΔL. Angular impulse equals the change in angular momentum.

What is the equation for rotational kinetic energy in terms of angular momentum?

Ek = L²/(2I)

How does rotational kinetic energy depend on angular velocity?

Ek is proportional to ω² when moment of inertia is constant.

How does angular momentum depend on moment of inertia?

L = Iω. For constant angular momentum, increasing I decreases ω.

What is the difference between torque and angular momentum?

Torque causes a change in angular momentum; angular momentum describes rotational motion.

What happens to angular velocity if moment of inertia doubles and angular momentum is conserved?

Angular velocity halves.

What happens to rotational kinetic energy if moment of inertia halves and angular momentum is conserved?

Rotational kinetic energy doubles.