Torque Calculator
Find the torque of a force from its size, the lever arm and the angle — or solve for any of them. Add several forces to find the net torque and whether a lever or seesaw balances, and calculate the force needed to balance it. A third mode gives the moment of inertia of common shapes and the angular acceleration a torque produces. Runs in your browser; free, no sign-up.
Torque is the turning effect of a force. It grows with the distance r from the pivot and with the part of the force at right angles to the arm: θ is the angle between the arm and the force, so 90° gives the most torque and 0° gives none.
Calculated
Between 0° and 180°. A push at right angles to the arm is 90°.
Results
32.48 N·m
Torque τ
The working
- Start from τ = r × F × sin(θ).
- τ = 0.25 × 150 × sin(60°) = 32.48 N·m
The same torque in other units
| Unit | Torque |
|---|---|
| N·m | 32.48 |
| lbf·ft | 23.95 |
| lbf·in | 287.4 |
| N·cm | 3248 |
| kN·m | 0.03248 |
How to Use the Torque Calculator
- 1. Choose the calculation. Torque of one force for τ = rF sin θ; Net torque and balance for a lever or seesaw with several forces; Moment of inertia for a shape and the angular acceleration a torque gives it.
- 2. Enter the values. For a single torque, pick the unknown (torque, distance, force or angle). For a lever, list each force with its distance from the pivot and whether it turns the object clockwise or counter-clockwise. Choose kgf to enter the weight of a mass in kilograms.
- 3. Read the result. The calculator shows the torque in your unit and in lbf·ft, lbf·in and N·cm, or the net torque, or the moment of inertia.
- 4. Follow the working. Every step is written with your numbers, including the unit conversions.
The Formulas
| Quantity | Formula | Notes |
|---|---|---|
| Torque of a force | τ = r F sin θ | θ is the angle between the lever arm r and the force F |
| Rotational equilibrium | Στ = 0 | Clockwise torque equals counter-clockwise torque |
| Moment of inertia | I = Σ m r² | kg·m²; standard results for common shapes |
| Angular acceleration | α = τ ÷ I | The rotational form of F = ma |
| Parallel-axis theorem | I = I_centre + M d² | Shifts the axis a distance d from the centre |
Reference: OpenStax University Physics, Volume 1, sections 10.5 to 10.7. One lbf·ft is 1.3558 N·m.
Torque measures how effectively a force turns an object about a pivot. The same push does more with a longer wrench, and most when it is at right angles to the arm (sin 90° = 1). Pushing along the arm (θ = 0° or 180°) gives no torque at all. Torque has the units newton-metre, the same dimensions as the joule, but it is not an energy.
Worked Examples
A wrench. A 150 N pull on a 0.25 m wrench, at 60° to the handle, gives τ = 0.25 × 150 × sin 60° = 32.48 N·m, which is 23.95 lbf·ft (the calculator's default). Pulling at right angles would give 37.5 N·m.
A seesaw. A 25 kg child sits 1.8 m from the pivot and a 60 kg adult sits 0.75 m on the other side. Using weights (kgf), the first gives 25 × 1.8 = 45 kgf·m and the second 60 × 0.75 = 45 kgf·m. In newton-metres both are 441.3 N·m, so the net torque is zero and the seesaw balances. The lighter child needs the longer arm. If one force were missing, the calculator gives the force needed at a distance you choose, or the distance needed for a force.
A flywheel. A solid disk of mass 10 kg and radius 0.2 m has I = ½ × 10 × 0.2² = 0.2 kg·m². A torque of 5 N·m gives it an angular acceleration of 5 ÷ 0.2 = 25 rad/s².
Moment of Inertia of Common Shapes
| Shape and axis | Formula | M = 10 kg, R = 0.2 m |
|---|---|---|
| Solid cylinder or disk, about its axis | ½ M R² | 0.2 kg·m² |
| Thin hoop or hollow cylinder, about its axis | M R² | 0.4 kg·m² |
| Solid sphere, about a diameter | ⅖ M R² | 0.16 kg·m² |
| Thin spherical shell, about a diameter | ⅔ M R² | 0.2667 kg·m² |
| Thin rod, about its centre | M L² ÷ 12 | — |
| Thin rod, about one end | M L² ÷ 3 | — |
| Rectangular plate, about its centre (perpendicular) | M (a² + b²) ÷ 12 | — |
For the same mass and radius, mass farther from the axis gives a larger I: the hoop has more than the disk, and the shell more than the solid sphere. Standard results; see OpenStax University Physics, Volume 1, section 10.5.
Solving Balance Problems
Choose the pivot where an unknown force acts. That force has zero lever arm and drops out of the torque sum, so you can solve for the others directly. The calculator treats the pivot as fixed and uses its own sign convention — counter-clockwise positive — so you only need to say which way each force turns the object. For a rigid body to stay at rest the forces must also sum to zero, which the Force Calculator can check.
Frequently Asked Questions
- What is the formula for torque?
- τ = r F sin θ, where r is the distance from the pivot to where the force acts, F is the force, and θ is the angle between the lever arm and the force. For a force at right angles to the arm this is simply τ = rF.
- What is the unit of torque?
- The newton-metre (N·m) in SI. It is dimensionally the same as a joule, but torque is not energy, so the unit is never written as joules. In imperial units torque is given in pound-feet (lbf·ft); 1 lbf·ft equals 1.3558 N·m.
- Why does the angle matter?
- Only the component of the force perpendicular to the lever arm turns the object, and that component is F sin θ. A force pointing straight at or away from the pivot has no perpendicular component, so it gives no torque however large it is.
- What is rotational equilibrium?
- An object is in rotational equilibrium when the net torque on it is zero: the clockwise torques equal the counter-clockwise torques. A balanced seesaw is an example. Combined with zero net force, the object stays at rest.
- What is moment of inertia?
- It is a measure of how hard an object is to start or stop turning, the rotational counterpart of mass. It depends on the mass and on how far it is from the axis: I = Σ m r². The same object has different moments of inertia about different axes.
- How are torque and power related?
- The power delivered by a torque on a shaft turning at angular speed ω is P = τ ω. A motor producing 20 N·m at 100 rad/s therefore delivers 2,000 W. The Work and Power Calculator converts that to horsepower.