Torque Calculator
Calculate torque as force times lever arm times the sine of the angle between them, from the applied force, radius, and angle in degrees.
Torque Calculator
Calculator
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Formula: τ = r × F × sin(θ)
Worked example — 25 N·m
Formula
τ = r × F × sin(θ)
Rotational force. Torque equals the cross product of position vector and force.
Worked Examples
Example 1: Wrench
Problem:50 N force, 0.5 m handle, 90°
Solution:τ=0.5×50×sin(90°)=25 N·m
Result:25 N·m
Frequently Asked Questions
What is torque in physics?
Torque (τ) is the rotational equivalent of force. It measures how effectively a force causes an object to rotate about an axis: τ = r × F × sin(θ), where r is the lever arm length, F is the applied force, and θ is the angle between r and F. The SI unit is Newton-metres (N·m).
How do you calculate torque with an angle?
Use τ = r × F × sin(θ). Maximum torque occurs at θ = 90° (sin 90° = 1). Example: a 50 N force applied at 0.4 m from the pivot at 60°: τ = 0.4 × 50 × sin(60°) = 0.4 × 50 × 0.866 ≈ 17.3 N·m.
What units are used for torque calculations?
Force F is in Newtons (N), lever arm r in metres (m), and angle θ in degrees (converted internally to radians). Torque τ is in Newton-metres (N·m). In imperial units, 1 N·m = 0.7376 ft·lbf. Automotive specs often use N·m or lb·ft.
What is the relationship between torque and power?
Power equals torque multiplied by angular velocity: P = τω, where ω is in radians per second. In practical terms: Power (W) = Torque (N·m) × 2π × RPM / 60. For cars: HP ≈ Torque (lb·ft) × RPM / 5252. A 300 lb·ft engine at 4000 RPM produces about 228 HP.
What is the difference between torque and work if both are measured in newton-metres?
Torque is a cross product, τ = r × F, and its direction lies along the rotation axis, while work is a dot product, W = F·d, and is a plain scalar. Because the vectors combine differently the two quantities are not interchangeable, so work is reported in joules and torque is kept in newton-metres even though the base units match. A bolt held at 40 N·m that never turns absorbs zero joules of work.
Why does torque fall to zero when the force is applied along the lever arm?
The sin(θ) term in τ = r × F × sin(θ) selects only the force component perpendicular to the lever arm. At θ = 0° or 180° the force pushes or pulls straight through the pivot, sin(θ) = 0, and no rotation results. That is why hauling a wrench directly toward the bolt achieves nothing, and why the handle should be swung at a right angle to the shaft for maximum effect.
What is the moment arm and how does it differ from the lever arm length r?
The moment arm is the perpendicular distance from the pivot to the force's line of action, equal to r × sin(θ). The lever arm r is the full straight-line distance to the point where the force acts. With r = 0.5 m and θ = 30°, the moment arm is 0.5 × 0.5 = 0.25 m, so a 50 N force yields 12.5 N·m instead of the 25 N·m it would give at 90°.
How does torque relate to angular acceleration and moment of inertia?
Newton's second law for rotation states that net torque = I × α, where I is the moment of inertia in kg·m² and α is angular acceleration in rad/s². A solid disc of mass 2 kg and radius 0.1 m has I = ½ × m × r² = 0.01 kg·m², so a steady 0.5 N·m torque produces α = 0.5 / 0.01 = 50 rad/s². Greater inertia means the same torque spins the body up more slowly.
How does an in-line extension bar change the torque a wrench actually delivers?
An extension fitted in line with the handle lengthens the effective lever arm from L to L + E, so the fastener receives more torque than the wrench dial reads. To hit a target value, set the wrench to target × L / (L + E). With a 0.45 m wrench, a 0.15 m in-line extension and a 100 N·m target, dial in 100 × 0.45 / 0.60 = 75 N·m. A crowfoot turned 90° to the handle changes nothing.
How do you balance torques on a lever in static equilibrium?
Pick a pivot, treat one rotation sense as positive (counter-clockwise by convention), then set the sum of torques to zero alongside the sum of forces. On a seesaw a 300 N child 1.0 m from the fulcrum creates 300 N·m, so a 200 N child balances it at 300 / 200 = 1.5 m on the opposite side. Choosing the pivot at an unknown reaction force removes that force from the torque equation.
Background & Theory
Torque at a Glance: Typical Values
Torque (τ) measures a force's tendency to rotate an object about a pivot. Because it depends on both the magnitude of the force and the angle at which it's applied, the same push can produce very different torque depending on where and how it's applied — pushing a door near its hinge takes much more force than pushing near the handle for the same rotating effect.
| Application | Typical torque |
|---|---|
| Car wheel lug nut | 80-140 N·m (varies by vehicle, check manufacturer spec) |
| Spark plug | 20-30 N·m |
| Small electronics screw (M2-M3) | 0.2-0.6 N·m |
| Passenger car engine (peak) | 150-400 N·m |
| Large diesel truck engine (peak) | 1,000-2,500 N·m |
Always torque fasteners to the manufacturer's exact specification with a calibrated torque wrench — over-tightening can strip threads or warp components, while under-tightening risks parts working loose.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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