How to Calculate Torque: Rotational Physics Guide
Learn torque τ = rF sin θ, lever arms, moments of force, and rotational equilibrium. Worked examples cover wrenches, doors, and see-saws, plus common mistakes and the Torque Calculator.
How to Calculate Torque
Torque (also called the moment of a force) measures how effectively a force tends to rotate an object about an axis or pivot. Pushing a door near the handle opens it easily; pushing near the hinge barely moves it — same force, different torque.
Torque is the rotational analogue of force. Force changes linear momentum; torque changes angular momentum. Engineers size bolts, motors, and shafts in newton-metres (N·m); mechanics talk about "foot-pounds" for the same idea.
The core formula
\tau = r F \sinθ
Where:
- \tau (tau) is torque in N·m
- r is the distance from the pivot (axis) to the point where the force is applied, in metres
- F is the magnitude of the force in newtons
- θ is the angle between the position vector (from pivot to application point) and the force vector
In vector form:
\vec\tau = \vecr × \vecF
Magnitude is rF\sinθ; direction is perpendicular to the plane of \vecr and \vecF (right-hand rule).
Lever arm and perpendicular force
Two equivalent views help intuition:
1. Perpendicular component of force. Only F_\perp = F\sinθ produces torque:
\tau = r F_\perp
A force aimed straight at the pivot (θ = 0 or 180°) produces zero torque — it cannot twist.
2. Moment arm (lever arm). The perpendicular distance from the pivot's line to the line of action of the force is \ell = r\sinθ:
\tau = F \ell
Lengthening the wrench handle increases \ell and therefore \tau for the same hand force.
Sign convention and equilibrium
In planar problems, assign:
- Positive torque for one sense (often counterclockwise)
- Negative for the opposite sense
For an object in rotational equilibrium:
Σ \tau = 0
Combined with Σ F_x = 0 and Σ F_y = 0, this is the condition for statics (ladders, beams, see-saws).
Worked example: tightening a bolt
Problem. You apply 80 N at the end of a 0.25 m wrench. The force is perpendicular to the wrench. Find the torque on the bolt.
θ = 90°, \quad \sin 90° = 1
\tau = (0.25)(80)(1) = 20 \text N·m
If instead you push at 60° to the wrench:
\tau = (0.25)(80)\sin 60° = 20 × 0.866 = 17.3 \text N·m
About 13% less — angle matters.
Worked example: opening a door
Problem. A door is 0.90 m wide. You push with 40 N perpendicular to the door, 0.80 m from the hinges. What torque do the hinges feel about the hinge axis?
\tau = (0.80)(40)(1) = 32 \text N·m
Push at only 0.20 m from the hinges with the same force:
\tau = (0.20)(40) = 8 \text N·m
Four times closer to the pivot means one-quarter the torque — which is why hinge-side pushes feel useless.
Worked example: balanced see-saw
Problem. Child A (30 kg) sits 2.0 m left of the fulcrum. Where must child B (40 kg) sit on the right for balance? Take g = 9.81\,\mathrmm/s^2.
Weights: W_A = 30g, W_B = 40g. Torques about the fulcrum (magnitudes):
\tau_A = (2.0)(30g), \quad \tau_B = d(40g)
Set equal for equilibrium:
2.0 × 30 = 40d \implies d = (60)/(40) = 1.5 \text m
Child B sits 1.5 m to the right. Mass cancelled g — balance depends on m × r, the moment of weight.
Torque, moment of inertia, and angular acceleration
Newton's second law for rotation about a fixed axis:
Σ \tau = Iα
Where I is the moment of inertia (kg·m²) and α is angular acceleration (rad/s²). Larger I (mass farther from the axis) needs more torque for the same α.
Example. A wheel with I = 0.50\,\mathrmkg· m^2 needs α = 4.0\,\mathrmrad/s^2:
\tau = Iα = 0.50 × 4.0 = 2.0 \text N·m
Power and work with torque
Constant torque through angle θ (radians) does work:
W = \tau θ
Average power when angular speed is \omega (rad/s):
P = \tau \omega
Motor nameplates often quote torque and RPM; convert RPM to rad/s with \omega = 2π n/60 before using P = \tau\omega.
Couples and pure moments
Two equal, opposite forces separated by distance d, with parallel lines of action, form a couple. Net force is zero, but net torque is:
\tau = Fd
(independent of the reference point). Steering wheels and turning a screwdriver rely on couples.
Units and conversions
Do not confuse N·m of torque with joules of energy: they share dimensions, but torque is not energy until multiplied by an angle in radians (W = \tauθ).
Applications
Automotive. Engine torque at the crankshaft, multiplied by gear ratios, becomes wheel torque. Low gears trade speed for torque to start the car.
Structural engineering. Beam bending moments are torques from loads about a section — design against excessive moment.
Robotics and prosthetics. Joint motors are specified by stall torque and continuous torque.
Sports. Bat and racket length increase tip speed and can increase torque about the wrists; grip strength and moment of inertia trade off.
Fasteners. Torque wrenches apply a target \tau so bolt tension stays in a safe range (friction at the threads makes the \tau–tension link approximate).
Common mistakes
Using degrees in W = \tauθ without converting. Angle must be in radians for work and for α in \tau = Iα.
Forgetting \sinθ. Parallel-to-radius forces do not produce torque.
Measuring r to the wrong point. r is from the chosen axis to where the force acts, not necessarily the object's centre of mass (unless that is your axis).
Adding forces instead of torques in statics. Equilibrium needs both Σ F = 0 and Σ\tau = 0.
Mixing lb and lbf carelessly in US customary units. Mass and force are easy to confuse; prefer SI for calculation, convert at the end.
Assuming the largest force always wins. A smaller force with a longer lever arm can dominate.
Frequently asked questions
Is torque a vector? Yes. In three dimensions it has direction along the axis of would-be rotation. In many classroom problems you only track signed magnitude in a plane.
Why do longer wrenches help? They increase r (or \ell), so the same hand force produces larger \tau. They also change your posture — use care so the tool does not slip.
What is the difference between torque and moment? In practice, often the same. "Bending moment" in beams is the internal torque about a section. Some texts reserve "torque" for twisting a shaft and "moment" for bending — the math is identical.
Can torque exist if the object does not rotate? Yes. Static torque can be balanced by another torque (held bolt, see-saw at rest). Net torque is zero; individual torques need not be.
How is horsepower related to torque? In imperial units a common rule is \mathrmhp = \tau_\mathrmft· lbf × \mathrmRPM / 5252. In SI, use P = \tau\omega with \omega in rad/s.
Does mass appear in \tau = rF\sinθ? Not directly. Mass (through I) appears when you ask how large α will be for a given net torque.
Summary
Torque is:
\tau = r F \sinθ = F \ell
Maximize the lever arm and apply force perpendicular to the radius for maximum twist. For statics, require Σ\tau = 0; for dynamics about a fixed axis, use Σ\tau = Iα. Keep angles and units consistent, and remember that force through the pivot produces no torque.
Solve torque problems with our Torque Calculator.