Wrench force at an angle
Known: F = 200 N, r = 0.30 m, θ = 30°.
Formula: τ = rF sinθ.
τ = (0.30 m)(200 N)sin(30°) = 30 N·m
Torque & Moment of Inertia Calculator (τ = Iα)
Calculate torque, mass moment of inertia, or angular acceleration from any two known values. You can also solve force torque with an angle, find inertia for common shapes, and analyze several signed torques.
Scope: this page calculates mass moment of inertia for rotational dynamics (units of mass × length²), not the area moment of inertia used in beam bending (units of length⁴). Results update when you choose Calculate.
Calculator inputs
Solve τnet = Iα
Choose the unknown, then enter the other two quantities. Positive is counterclockwise; negative is clockwise.
A signed torque is allowed.
Must be greater than zero.
A signed acceleration is allowed.
Optional energy and power
Zero returns 0 J and 0 W.
Solve τ = rF sinθ
Use the actual radius and angle, or switch to the already-perpendicular lever arm r⊥.
Distance from axis to force application.
Principal angle from 0° to 180°.
Advanced: multiple forces and net torque
Each contribution is τᵢ = ±rᵢFᵢsinθᵢ. Add friction or drag as an opposing direction.
Mass moment of inertia from shape
Select the body and the named rotation axis. The diagram and required dimensions update together.
Available for centroidal base-axis formulas. Point-mass and rod-end cases already specify non-centroidal axes.
Solution
Choose a mode, enter the known values, and select Calculate.
Answer
—
Perpendicular component
Rotational kinetic energy
K = ½Iω²
Rotational power
P = τω
Calculation steps
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Quick answer and how to use the calculator
For fixed-axis rotational dynamics, use τnet = Iα. For a single force, use τ = rF sinθ = r⊥F. Torque is created only by the force component perpendicular to the radius.
- Choose Rotational Dynamics, Force Torque, or Shape Inertia.
- Select the unknown or body/axis, enter the requested values, and choose units independently.
- Select Calculate to see conversions, substitutions, arithmetic, and rounded and unrounded answers.
- For several forces, open the advanced panel and assign each contribution a clockwise or counterclockwise sign.
Worked examples
Disk accelerated by known torque
Known: m = 4 kg, R = 0.25 m, τ = 2.5 N·m.
Formulas: I = ½mR² and α = τ/I.
I = 0.125 kg·m²
α = 2.5/0.125 = 20 rad/s²
Flywheel energy and power
Known: I = 12 kg·m², ω = 20 rad/s, τ = 15 N·m.
Formulas: K = ½Iω² and P = τω.
K = ½(12)(20²) = 2,400 J
P = (15)(20) = 300 W
Formula reference and variable definitions
| Formula | Use it when | Notes |
|---|---|---|
| τ = rF sinθ | One force acts at radius r. | θ is the angle between the radius vector and force. Equivalent to rF⊥ or r⊥F. |
| τnet = Στ = Iα | Finding a rigid body's angular acceleration or one unknown dynamics variable. | τnet is the signed sum of external torques about the selected fixed axis. |
| K = ½Iω² | Finding rotational kinetic energy at angular speed ω. | ω must be in rad/s for direct SI calculation. |
| P = τω | Torque and angular velocity are collinear and instantaneous power is required. | Signed P indicates whether torque adds or removes rotational energy. |
| L = Iω | A rigid body rotates about a principal fixed axis. | L is angular momentum in kg·m²/s. |
| I = Ic + md² | The required axis is parallel to a center-of-mass axis. | d is the perpendicular separation between the two axes. |
Variables
τ: torque; I: mass moment of inertia; α: angular acceleration; r: radius vector magnitude; F: force; θ: angle between r and F; ω: angular speed.
Signs and radians
This calculator uses counterclockwise as positive and clockwise as negative. A radian is the dimensionless ratio of arc length to radius; trigonometric calculations convert degree inputs to radians internally.
Units
Internal calculations use N, m, kg, seconds, radians, N·m, and kg·m². Mixed input units are converted before arithmetic. A newton-metre of torque should not be labelled a joule.
Why the axis matters
Mass farther from the axis contributes more because I = ∫r²dm. Always match the formula's axis to the real rotation axis; shape alone is not enough.
Assumptions and common mistakes
Assumptions
- Rigid body with a fixed axis.
- Scalar planar torque; the selected sign represents the right-hand-rule direction.
- Shape formulas assume uniform mass density and the axis named in the selector.
- τ = Iα uses the net external torque and constant I.
Common errors
- Using the full force instead of F sinθ.
- Using the wrong angle: θ is between r and F.
- Mixing centimetres, inches, grams, or pounds without conversion.
- Adding opposing torque magnitudes instead of subtracting signed contributions.
- Confusing mass moment of inertia with area moment of inertia.
Opposing torque
Friction, drag, a brake, or another force may oppose the applied torque. Include each as a signed contribution. Static equilibrium occurs when Στ = 0; a zero net torque also means α = 0 for finite I.
Torque and moment of inertia FAQs
How are torque, moment of inertia, and angular acceleration related?
For a rigid body about a fixed axis, net torque equals mass moment of inertia times angular acceleration: τnet = Iα.
How do I solve τ = Iα for each variable?
Use τ = Iα for torque, I = τ/α for moment of inertia, or α = τ/I for angular acceleration.
What is the difference between mass and area moment of inertia?
Mass moment of inertia (kg·m²) describes resistance to angular acceleration. Area moment of inertia (length⁴) describes a cross-section's resistance to bending.
Why does the rotation axis matter?
I depends on every mass element's squared distance from the chosen axis, so the same object has different values about different axes.
When is sinθ required in a torque calculation?
Use sinθ when force is not perpendicular to the radius. Only the perpendicular component F sinθ creates torque.
Is a newton-metre the same as a joule?
They share SI base dimensions, but torque and energy are different physical quantities. Write torque as N·m, not J.
Can torque be negative?
Yes. Here, counterclockwise torque is positive and clockwise torque is negative.
How does the parallel-axis theorem work?
For an axis parallel to one through the centre of mass, I = Ic + md², where d is the perpendicular distance between axes.
What happens when friction creates opposing torque?
Give friction torque the opposite sign and include it in Στ. Then α = Στ/I.
Methodology, references, and checked cases
Prepared and technically reviewed by: Starlight Robotics editorial team. No individual professional credential is claimed.
Last reviewed: 15 July 2026.
Rounding: calculations retain JavaScript double precision; displays use up to 8 significant digits, and the steps include an unrounded machine value when rounding changes it.
Conversion constants: 1 lbf = 4.4482216152605 N; 1 in = 0.0254 m; 1 ft = 0.3048 m; 1 lb = 0.45359237 kg. Length and pound conversions are treated as exact for this tool.
Formula references: OpenStax: Torque, OpenStax: Newton's second law for rotation, OpenStax: moments of inertia and the parallel-axis theorem, and NIST SP 811: SI and conversion factors.
Reference case 1
r = 0.30 m, F = 200 N, θ = 30° → τ = 30 N·m.
Reference case 2
Solid disk, m = 4 kg, R = 0.25 m → I = 0.125 kg·m².
Reference case 3
τ = 2.5 N·m and I = 0.125 kg·m² → α = 20 rad/s².