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The rotational equations of motion

rotational kinematicsangular motion formulastorque and angular momentumrotational dynamicsmoment of inertia formulas

The angular twins of the linear equations — θ, ω and α in place of d, v and a, and torque, moment of inertia and Iω in place of F, m and p.

Angular Velocity (ω = θ/t)

ω=θt\omega = \frac{\theta}{t}

Average angular velocity: the angle swept divided by the time taken.

Angular Acceleration

α=ω−ω0t\alpha = \frac{\omega - \omega_0}{t}

Average angular acceleration: the change in angular velocity divided by the time taken.

Angular Displacement (θ = ω₀t + ½αt²)

θ=ω0t+12αt2\theta = \omega_0 t + \tfrac{1}{2} \alpha t^{2}

Angle turned under constant angular acceleration, the rotational twin of x = v₀t + ½at².

Linear Speed from Rotation (v = ωr)

v=ωrv = \omega r

A point at radius r on a rotating body moves with linear speed ωr.

Torque

τ=rFsin⁡θ\tau = r F \sin\theta

Turning effect of a force applied at distance r from a pivot, at angle θ to the lever arm.

Torque with a Lever Arm (τ = rF sin θ)

τ=rFsin⁡θ\tau = r F \sin\theta

Torque produced by a force applied at distance r from the pivot, at angle θ to the lever.

Moment of Inertia: Point Mass

I=mr2I = m r^{2}

Rotational inertia of a compact mass circling at radius r from the axis.

Moment of Inertia: Solid Disk

I=12mr2I = \tfrac{1}{2} m r^{2}

Rotational inertia of a uniform solid disk or cylinder about its central axis.

Moment of Inertia: Solid Sphere

I=25mr2I = \tfrac{2}{5} m r^{2}

Rotational inertia of a uniform solid sphere about an axis through its center.

Newton's Second Law for Rotation (τ = Iα)

τ=Iα\tau = I \alpha

Net torque equals moment of inertia times angular acceleration — F = ma for spinning things.

Rotational Kinetic Energy

KErot=12Iω2KE_{rot} = \tfrac{1}{2} I \omega^{2}

Kinetic energy stored in rotation: half the moment of inertia times angular velocity squared.

Angular Momentum (L = Iω)

L=IωL = I \omega

Angular momentum of a rotating body: moment of inertia times angular velocity.

Rotational Power (P = τω)

P=τωP = \tau \omega

Mechanical power delivered by a torque turning at angular velocity ω.

How they fit together

Rotation is not a separate subject; it is linear mechanics with every quantity swapped for its angular counterpart. Displacement becomes angle, velocity becomes ω, acceleration becomes α, force becomes torque, mass becomes moment of inertia, and momentum becomes Iω. F = ma turns into τ = Iα, ½mv² turns into ½Iω², and P = Fv turns into P = τω. Learn the mapping and you already know these formulas.

The one genuinely new idea is moment of inertia, and it is where the mistakes live. Mass is a single number for an object; rotational inertia is not, because it depends on where the mass sits relative to the axis and on which axis you chose. A hoop and a disk of identical mass and radius have different I, which is why they lose a race down a ramp by different margins. So pick the moment-of-inertia formula that matches the actual shape and the actual axis before touching anything else. After that, keep every angle in radians — v = ωr and the arc relations are only true in radians — and remember that torque needs the perpendicular distance, so pushing a wrench along its own handle achieves nothing at all.