Mechanicsτ = Iα

Rotational Motion Simulator

Explore torque, angular velocity, and moment of inertia. Apply a torque to a disk and see angular acceleration in real time.

Parameters

Torque τ

N·m

Disk mass m

Radius r

Derived

I = ½mr²0.1250 kg·m²

α = τ/I40.00 rad/s²

I0.0000 kg·m²

τ5.0 N·m

α0.000 rad/s²

ω0.000 rad/s

θ0.00 rad

KE_rot0.000 J

v_rim0.000 m/s

About Rotational Motion

Rotational motion is the angular analogue of linear motion. Instead of force, mass, and linear acceleration, we have torque, moment of inertia, and angular acceleration. The governing equation τ = Iα mirrors F = ma.

ℹEvery linear kinematic quantity has a rotational analogue: displacement → angle θ, velocity → angular velocity ω, acceleration → angular acceleration α, mass → moment of inertia I, force → torque τ.

Key Variables

Symbol	Name	Unit	Description
τ	Torque	N·m	Rotational force: τ = r × F
I	Moment of Inertia	kg·m²	Rotational inertia: I = ½mr² for a solid disk
α	Angular Acceleration	rad/s²	α = τ/I
ω	Angular Velocity	rad/s	Rate of change of angle: ω = ω₀ + αt
θ	Angle	rad	Total rotation: θ = ½αt² (from rest)
KE_rot	Rotational KE	J	KE_rot = ½Iω²
m	Disk Mass	kg	Total mass of the disk
r	Disk Radius	m	Radius of the disk

Worked Example

A disk of mass m = 2 kg, radius r = 0.5 m has torque τ = 5 N·m applied. Find α, and ω after 3 s.

I = \frac{1}{2} m r^{2} = \frac{1}{2} (2) (0.5)^{2} = 0.25 kg \cdotp m^{2}

α = \frac{τ}{I} = \frac{5}{0.25} = 20 rad/s^{2}

ω (3) = 0 + α t = 20 \times 3 = 60 rad/s

K E_{rot} = \frac{1}{2} I ω^{2} = \frac{1}{2} (0.25) (60)^{2} = 450 J

Effect of Radius on Moment of Inertia

For a solid disk, I = ½mr². Doubling the radius quadruples I, making it much harder to angularly accelerate. This is why large heavy wheels are hard to spin up but also hard to stop once rotating.

💡Increase the radius in the simulator — notice how α decreases and ω builds up more slowly even with the same torque. This illustrates why figure skaters pull their arms in to spin faster (conserving angular momentum).

Rotational Analogues

Linear: F = ma ↔ Rotational: τ = Iα
Linear: KE = ½mv² ↔ Rotational: KE = ½Iω²
Linear: p = mv ↔ Rotational: L = Iω (angular momentum)
Linear: x = ½at² ↔ Rotational: θ = ½αt²

Rotational Motion Formulas

Newton's Second Law for Rotation

τ = I α ⟹ α = \frac{τ}{I}

Moment of Inertia (Solid Disk)

I = \frac{1}{2} m r^{2}

Angular Kinematics

ω (t) = ω_{0} + α t θ (t) = θ_{0} + ω_{0} t + \frac{1}{2} α t^{2}

Rotational Kinetic Energy

K E_{rot} = \frac{1}{2} I ω^{2}

Angular Momentum

L = I ω (conserved when τ_{net} = 0)

Rotational Formula	Linear Analogue	Notes
τ = Iα	F = ma	τ in N·m, I in kg·m², α in rad/s²
I = ½mr²	m (inertia)	For solid disk; differs for other shapes
ω = ω₀ + αt	v = v₀ + at	Angular velocity from constant α
θ = ½αt²	x = ½at²	Angle from rest with constant α
KE = ½Iω²	KE = ½mv²	Rotational kinetic energy
L = Iω	p = mv	Angular momentum

ℹMoment of inertia depends on both mass and how that mass is distributed relative to the rotation axis. I = ½mr² for a solid disk, but I = mr² for a thin ring where all mass is at the rim.

Frequently Asked Questions

Why is moment of inertia different for different shapes?

Moment of inertia I = ∫r² dm depends on how far each piece of mass is from the rotation axis. For a solid disk, half the mass is close to the centre (small r), giving I = ½mr². For a ring, all mass is at radius r, giving I = mr² — twice as much rotational inertia for the same mass and radius.

What is torque?

Torque is the rotational equivalent of force. τ = r × F, where r is the distance from the rotation axis to where the force is applied, and F is the perpendicular component of that force. A larger moment arm or larger force creates a larger torque.

Why do figure skaters spin faster when they pull their arms in?

When no external torque acts (free spin), angular momentum L = Iω is conserved. Pulling arms in reduces I (mass closer to axis). Since L = Iω is constant, ω must increase — the skater spins faster.

What happens to rotational KE if I double the torque?

Doubling torque doubles angular acceleration α. After time t, ω = αt doubles, and KE = ½Iω² quadruples (since KE scales as ω²). More torque means energy builds up much faster.

Is angular momentum the same as torque?

No. Angular momentum L = Iω is the rotational state of motion (like linear momentum p = mv). Torque τ = dL/dt is the rate of change of angular momentum (like force F = dp/dt). Torque causes changes in angular momentum.