Torque is one of the most important quantities in rotational mechanics — it governs everything from loosening a bolt to the power output of a car engine. This calculator covers five core torque relationships and lets you solve for any variable, displaying the formula used and every step of the arithmetic so you can check your understanding or catch mistakes in your working.
What this calculator covers
The tool is organised into five tabs, each targeting a different torque formula:
- Basic Torque (τ = F·r·sinθ) — the defining equation. Given any three of torque, force, moment arm, and angle, solve for the fourth.
- Newton’s 2nd for Rotation (τ = Iα) — just as F = ma in linear dynamics, net torque equals moment of inertia times angular acceleration. Solve for any one of τ, I, or α.
- Rotational Power (P = τω) — power delivered by a rotating shaft equals torque times angular velocity. Enter RPM or rad/s; the calculator converts automatically.
- Rotational Work (W = τθ) — energy transferred by a constant torque over an angular displacement. Angles are entered in degrees; the calculation converts to radians internally.
- Lever / Moment Balance (F_e·d_e = F_l·d_l) — the principle of moments for a simple first-class lever, plus mechanical advantage, velocity ratio, and efficiency.
How the physics works
Basic torque
The moment of a force about a pivot is defined as:
τ = F · r · sin(θ)
where F is the applied force in newtons, r is the perpendicular distance (moment arm) from the pivot to the line of action in metres, and θ is the angle between the force vector and the lever arm. The SI unit is the newton-metre (N·m). Maximum torque occurs when force is applied at right angles (θ = 90°, sin = 1); pushing along the arm (θ = 0°) produces zero torque.
Newton’s second law for rotation
Linear Newton’s second law F = ma has a direct rotational counterpart:
τ = I · α
where I is the moment of inertia (kg·m²) — the rotational analogue of mass — and α is angular acceleration in rad/s². The calculator includes a reference table of I for common geometries (disk, sphere, rod, etc.) so you can look up the right value.
Rotational power
A shaft transmitting torque τ at angular velocity ω delivers power:
P = τ · ω
with P in watts (W). Because ω = 2π × n/60 when n is in RPM, the power tab lets you enter revolutions per minute directly. A 50 N·m torque at 3,000 RPM (≈ 314 rad/s) produces about 15.7 kW — exactly the kind of calculation needed for motor and engine problems.
Rotational work
When a constant torque turns a body through angle θ (in radians), the work done is:
W = τ · θ
The tab accepts θ in degrees for convenience, converting to radians (multiply by π/180) before computing. One full revolution (360°) equals 2π radians, so a 30 N·m torque completing one full turn does W = 30 × 2π ≈ 188.5 J.
Lever and moment balance
For a simple lever in equilibrium about a fulcrum, the principle of moments gives:
F_effort × d_effort = F_load × d_load
The Lever tab also reports:
- Mechanical advantage (MA) = F_load / F_effort — how much the lever multiplies your force.
- Velocity ratio (VR) = d_effort / d_load — the ratio of arm lengths.
- Efficiency = (MA / VR) × 100% — 100% for an ideal frictionless lever; real levers are slightly lower.
Worked example
A mechanic uses a spanner to tighten a bolt. The spanner handle is 0.25 m long and the mechanic pushes with 80 N at 90° to the handle.
Using τ = F · r · sin(θ):
τ = 80 N × 0.25 m × sin(90°) = 80 × 0.25 × 1 = 20 N·m
If the bolt’s fastener assembly has a moment of inertia of 0.004 kg·m² and the desired angular acceleration is 5,000 rad/s², then the required torque from Newton’s 2nd is:
τ = I · α = 0.004 × 5000 = 20 N·m
Both tabs agree — confirming the mechanic is applying exactly the right force. The calculation runs instantly in your browser without sending any data to a server.