What is the torque formula?

The torque formula is τ = r × F × sin(θ), where r is the lever arm length (distance from the pivot to the point where force is applied), F is the applied force, and θ is the angle between the force vector and the lever arm. Maximum torque occurs at θ = 90°.

Why does angle matter in torque calculations?

The sin(θ) term captures the effective component of force perpendicular to the lever arm. Only the perpendicular component creates rotation. At 90° (force perpendicular to arm), sin(90°) = 1 and torque is maximum. At 0° or 180° (force parallel to arm), sin = 0 and torque is zero.

How is torque related to angular acceleration?

The rotational Newton's Second Law states τ = I × α, where τ is net torque (N·m), I is moment of inertia (kg·m²), and α is angular acceleration (rad/s²). This is the rotational equivalent of F = ma. A larger torque on a given body produces greater angular acceleration.

What is the parallel-axis theorem?

The parallel-axis theorem states that the moment of inertia about any axis equals the moment of inertia about a parallel axis through the centre of mass plus M×d², where M is total mass and d is the distance between axes: I = I_cm + Md². It allows computing I for an axis offset from the centre of mass.

What units is torque measured in?

The SI unit of torque is Newton-metre (N·m). Other common units include: kN·m (kilonewton-metre), N·cm, lbf·ft (pound-force foot, 1 lbf·ft = 1.35582 N·m), and lbf·in (1 lbf·in = 0.11298 N·m). Note: N·m for torque and J (joules) for energy are dimensionally equivalent but physically distinct quantities.

Torque Calculator – τ = rF sinθ, Moment of Inertia & Rotation

Quick Presets

Solve For

Force F

Lever Arm r

Angle θ (degrees)

°

Torque τ

Basic torque inputs — same as Basic tab. Solve-for selector applies.

Select what to solve for, enter known values, get instant results.

What Is Torque?

Torque (also called moment of force) is the rotational equivalent of linear force. It measures how much a force causes an object to rotate about a pivot point or axis. When you tighten a bolt with a wrench, push a door open, or pedal a bicycle, you are applying torque.

The key insight: the same force produces more torque if applied farther from the pivot, or if applied perpendicularly. This is why a longer wrench makes it easier to loosen a stuck bolt.

τ = rF sinθ

Basic torque formula

τ = Iα

Rotational Newton's 2nd law

W = τθ

Rotational work (J)

The Torque Formula τ = rF sinθ

The torque produced by a force depends on three factors:

r — Lever arm: distance from the pivot to where the force is applied (metres)
F — Magnitude of the applied force (Newtons)
θ — Angle between the force vector and the lever arm direction

τ = r × F × sin(θ)

Maximum torque at θ = 90° (force perpendicular to arm)

When θ = 90°, sin(90°) = 1 and all the force contributes to rotation. When θ = 0° or 180°, the force is parallel to the arm and produces no rotation.

Moment of Inertia for Common Shapes

Moment of inertia I measures rotational inertia — an object's resistance to changes in rotation. It depends on both total mass and how that mass is distributed.

Shape	Formula
Solid disk / cylinder	I = ½MR²
Hoop / thin ring	I = MR²
Solid sphere	I = (2/5)MR²
Thin spherical shell	I = (2/3)MR²
Rod about center	I = (1/12)ML²
Rod about end	I = (1/3)ML²
Point mass	I = mr²

Rotational Newton's Second Law

Just as F = ma governs linear motion, τ = Iα governs rotational motion. Net torque equals moment of inertia times angular acceleration:

α = τ / I (rad/s²)

A 50 N·m torque on a solid disk with I = 0.9 kg·m² produces α = 50/0.9 ≈ 55.6 rad/s². The same torque on a hoop of equal mass produces less angular acceleration because its moment of inertia is higher.

Rotational Work and Power

Rotational Work

W = τ · θ

θ in radians, W in joules

Rotational Power

P = τ · ω

ω in rad/s, P in watts

These are the rotational analogues of W = Fd and P = Fv. A motor producing 100 N·m at 100 rad/s delivers 10,000 W = 10 kW of power.

Real-World Applications

Automotive Engines

Engine torque (N·m) determines pulling power; horsepower (kW) is torque × RPM. A diesel engine at 400 N·m and 2500 RPM produces ≈ 104 kW.

Fastener Tightening

Torque wrenches ensure bolts are tightened to spec — too little and joints loosen; too much and bolts shear. Critical for engines, wheels, and structural joints.

Bicycles and Gears

Gear ratios trade torque for speed. Low gears provide high torque (for hills); high gears reduce torque but increase speed. Rider force × crank length = torque.

Structural Engineering

Beams experience bending moments (torques) under load. Engineers calculate moment diagrams to ensure structures don't fail in rotation at critical cross-sections.

Worked Examples

Example 1 — Wrench Bolt (θ=90°)

Given: F=200 N, r=0.25 m, θ=90°

τ = rF sin(90°) = 0.25 × 200 × 1

τ = 50 N·m

Example 2 — Angled Wrench (θ=60°)

Given: F=200 N, r=0.25 m, θ=60°

τ = 0.25 × 200 × sin(60°)

= 50 × 0.866 = 43.3 N·m

Example 3 — Solid Disk Moment of Inertia

Given: M=20 kg, R=0.3 m (disk)

I = ½ × 20 × 0.3² = ½ × 20 × 0.09

I = 0.9 kg·m²

Example 4 — Angular Acceleration

Given: τ=50 N·m, I=0.9 kg·m²

α = τ/I = 50 / 0.9

α = 55.6 rad/s²

Frequently Asked Questions

Torque (τ) is the rotational equivalent of force. It measures how much a force acting on an object causes it to rotate about an axis. Torque depends on the magnitude of the force, the distance from the pivot (lever arm), and the angle between the force and the lever arm.

The torque formula is τ = r × F × sin(θ), where r is the lever arm length, F is the applied force, and θ is the angle between the force vector and the lever arm. Maximum torque occurs at θ = 90° (force perpendicular to arm).

The sin(θ) term captures the effective component of force perpendicular to the lever arm. Only the perpendicular component creates rotation. At 90°, sin(90°) = 1 and torque is maximum. At 0° or 180° (force parallel to arm), sin = 0 and torque is zero.

Moment of inertia (I) is a measure of an object's resistance to angular acceleration — the rotational analogue of mass. It depends on how mass is distributed relative to the rotation axis. A hollow ring has I = MR² while a solid disk has I = ½MR², because the ring's mass is all at the rim.

Force causes linear (translational) acceleration according to F = ma. Torque causes angular (rotational) acceleration according to τ = Iα. Force is measured in Newtons; torque in Newton-metres (N·m). A force applied at a distance from a pivot creates torque; the same force through the pivot creates no torque.

The rotational Newton's Second Law states τ = I × α, where τ is net torque (N·m), I is moment of inertia (kg·m²), and α is angular acceleration (rad/s²). A larger torque on a given body produces greater angular acceleration.

The parallel-axis theorem states that I = I_cm + Md², where I_cm is the moment of inertia about an axis through the centre of mass, M is total mass, and d is the distance between axes. It allows computing I for any axis parallel to the CM axis.

The SI unit of torque is Newton-metre (N·m). Other common units include kN·m, N·cm, lbf·ft (1 lbf·ft = 1.35582 N·m), and lbf·in. Note: N·m for torque and J (joules) for energy are dimensionally equivalent but physically distinct quantities.

Torque Calculator

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