Physics

Rotation

Angular velocity, moment of inertia, torque, angular momentum, rolling.

Basics

Angular velocity and the tangent

If a rigid body turns about a fixed axis, every point shares the same angular velocity ω, and tangential speed is v = r ω. With angular acceleration α, a_t = r α; centripetal acceleration is ω² r. Use radians or v = r ω fails. Fix a clockwise/counterclockwise sign per problem. θ, ω, α sit where x, v, a sit in translation.

Moment of inertia

Moment of inertia I = Σ m r² measures how far mass sits from the axis — rotational inertia. A thin hoop is MR², a disk about (1/2)MR², a solid sphere about (2/5)MR². The parallel-axis theorem I = I_cm + M d² shifts to an axis not through the centre. Larger I means smaller angular acceleration for the same torque.

Torque and angular momentum

Torque τ = r × F = I α (fixed axis, constant I). The rotational work analogue is τ θ; kinetic energy is (1/2) I ω². Angular momentum L = I ω about that axis; if net torque is 0, L is conserved. A skater pulling arms in drops I and raises ω. A gyroscope feels stubborn because L has a direction.

Rolling

Rolling without slip ties v = r ω between centre and contact. The contact point is instantaneously at rest, so static friction need not do work. Total energy is translational (1/2)Mv² plus rotational (1/2)I ω². From the same height, a larger I finishes slower. If it slips, kinetic friction removes energy.

Formulas

Tangential speed

v = r ω

ω in rad/s. One turn is 2π rad.

Symbols

r perpendicular distance to the axis
ω angular velocity

Parallel-axis theorem

I = I_cm + M d²

An axis parallel to one through the CM, distance d away.

Symbols

I_cm I about the centre of mass
d distance between the axes

Rotational kinetic energy

K = ½ I ω²

For rolling, add translational ½ M v².

Symbols

I moment of inertia about that axis

Angular momentum (axis)

L = I ω, τ_net = dL/dt

Net torque changes angular momentum.

Symbols

L angular momentum

Key table

Disk vs hoop	same M, R: hoop has larger I, slower down a hill
Right-hand axis	fingers curl with rotation; thumb is ω and L
Radians	a pure number. 90° = π/2 rad