Torque is force applied at a distance from a pivot point

Torque measures how much a force will cause something to rotate around a fixed point. The basic formula is torque equals force times distance: τ = F × r. The force must be perpendicular to the lever arm (the distance from the pivot), and the result is measured in newton-meters (N⋅m) or foot-pounds (ft⋅lb).

The distance that matters is not the total length of the object, but the perpendicular distance from the pivot point to where the force is applied. If you push straight down on a wrench 0.3 meters from the bolt, and you push with 100 newtons of force, the torque is 30 newton-meters. If you push at an angle instead of perpendicular, the effective torque is smaller because only part of your force contributes to rotation.

Torque appears in everyday work: tightening a bolt with a wrench, opening a door, pedaling a bicycle, or rotating a steering wheel. The longer the wrench, the more torque you produce with the same push. The farther from the hinge you push a door, the easier it swings open.

Key Takeaways

  • The torque formula is force times perpendicular distance from the pivot: τ = F × r.
  • Force must be perpendicular to the lever arm; if you push at an angle, only the perpendicular component counts.
  • Longer lever arms produce more torque with the same force, which is why long wrenches tighten bolts more easily than short ones.
  • Torque is measured in newton-meters (metric) or foot-pounds (imperial), depending on your units for force and distance.
  • Direction matters: torque can be clockwise or counterclockwise, and opposing torques can cancel each other out.

The three parts of the torque equation

The torque formula has three components: the force applied, the distance from the pivot, and the angle between them. Force is measured in newtons (N) in metric units or pounds-force (lbf) in imperial units. Distance (called the lever arm or moment arm) is measured in meters (m) or feet (ft). The angle between the force direction and the lever arm determines whether you use the full force or only part of it.

When the force is perpendicular to the lever arm, the angle is 90 degrees and you use the full force. The formula becomes straightforward: τ = F × r. When the force is not perpendicular, you multiply by the sine of the angle: τ = F × r × sin(θ). For example, if you push a wrench at a 45-degree angle instead of straight down, your effective torque is only about 71 percent of what it would be at 90 degrees, because sin(45°) ≈ 0.707.

In most practical situations, you explore force as close to perpendicular as possible to maximize torque. A wrench handle is designed so you push straight down on it. A door handle is positioned so you push horizontally, perpendicular to the hinge. Understanding this angle is why a wrench that is too short or held at the wrong angle requires more effort.

Step-by-step calculation with real numbers

Start by identifying three things: the force you are explore, the distance from the pivot to where the force is applied, and whether the force is perpendicular to the lever arm.

Example 1: Tightening a bolt with a wrench. You explore 150 newtons of force straight down on a wrench that is 0.4 meters long, measured from the bolt to where you push. The force is perpendicular to the wrench. Torque = 150 N × 0.4 m = 60 N⋅m.

Example 2: Opening a door. You push with 50 newtons of force on a door handle that is 0.9 meters from the hinge. The force is perpendicular to the door. Torque = 50 N × 0.9 m = 45 N⋅m. If you pushed at the hinge instead (0 meters away), the torque would be zero and the door would not open, no matter how hard you pushed.

Example 3: Pushing at an angle. You push on a wrench with 100 newtons of force, but you push at a 60-degree angle instead of straight down. The wrench is 0.5 meters long. Torque = 100 N × 0.5 m × sin(60°) = 100 × 0.5 × 0.866 = 43.3 N⋅m. The angled push produces less torque than a perpendicular push would.

Converting between metric and imperial units

Torque in metric units (newton-meters) and imperial units (foot-pounds) are not interchangeable without conversion. One newton-meter equals approximately 0.738 foot-pounds. One foot-pound equals approximately 1.356 newton-meters.

If a bolt specification calls for 50 foot-pounds of torque, convert to metric: 50 ft⋅lb × 1.356 = 67.8 N⋅m. If you have a torque wrench marked in newton-meters and need to tighten to 40 N⋅m, convert to imperial: 40 N⋅m ÷ 1.356 = 29.5 ft⋅lb. Most modern torque wrenches have both scales, so you can read the unit you need directly.

When working with force and distance in different units, convert both to the same system before multiplying. If force is in pounds-force and distance is in feet, the result is in foot-pounds. If force is in newtons and distance is in meters, the result is in newton-meters. Mixing units (like newtons and feet) produces a number that is not a standard torque unit and is difficult to use.

Why direction and opposing torques matter

Torque has direction: clockwise or counterclockwise. When two torques act on the same pivot, they can add together or cancel each other out. If you tighten a bolt (clockwise torque) with 60 N⋅m and someone else tries to loosen it (counterclockwise torque) with 40 N⋅m, the net torque is 20 N⋅m clockwise, and the bolt continues to tighten.

In balanced systems, opposing torques are equal and the object does not rotate. A seesaw is balanced when the child on one side produces the same torque as the child on the other side. A heavier child sitting closer to the pivot can balance a lighter child sitting farther away. If the heavier child weighs 400 newtons and sits 1 meter from the pivot, the torque is 400 N⋅m. A lighter child weighing 200 newtons must sit 2 meters away to produce the same 400 N⋅m torque and balance the seesaw.

In engineering and mechanics, calculating net torque (the sum of all torques acting on an object) tells you whether the object will rotate and how fast. If net torque is zero, the object is in rotational equilibrium and will not start spinning. If net torque is greater than zero, the object will accelerate in the direction of the larger torque.

Common mistakes when calculating torque

The most common error is using the wrong distance. Many people measure the total length of a wrench or lever instead of the perpendicular distance from the pivot to the point where force is applied. If a wrench is 0.5 meters long but you only push on the last 0.3 meters of it, the distance that matters is 0.3 meters, not 0.5 meters.

Another frequent mistake is ignoring the angle. If you push a wrench at a 30-degree angle instead of perpendicular, you cannot use the full force in the torque formula. You must multiply by sin(30°) = 0.5, which means your effective torque is only half what you calculated. Many people push wrenches at awkward angles and wonder why they need more effort than expected.

A third error is forgetting that torque depends on both force and distance. Doubling the force doubles the torque, but doubling the distance also doubles the torque. A longer wrench is as effective as pushing harder. This is why mechanics use long wrenches on stubborn bolts and why doors have handles far from the hinge.

Torque in motors, engines, and rotating equipment

In motors and engines, torque is the rotational force produced at the shaft. An electric motor rated at 10 N⋅m produces enough rotational force to tighten a bolt that requires 10 N⋅m of torque. A car engine produces torque at different speeds; the specification sheet lists peak torque (the maximum) and the engine speed (in revolutions per minute) at which that peak occurs.

Power and torque are related but different. Power is the rate at which work is done, measured in watts or horsepower. Torque is the rotational force at a single when ready. A motor can produce high torque at low speed or lower torque at high speed while delivering the same power. This is why a truck engine produces high torque at low RPM (good for pulling) and a sports car engine produces high torque at high RPM (good for speed).

When selecting a motor or engine for a task, you need to know the torque required at the speed you plan to operate. A winch that lifts a heavy load needs high torque at low speed. A fan that spins fast needs lower torque at high speed. The torque formula remains the same, but the practical process changes based on how the rotating equipment is used.

Frequently Asked Questions

What is the difference between torque and force?

Force is a push or pull in a straight line, measured in newtons or pounds. Torque is rotational force around a pivot point, measured in newton-meters or foot-pounds. You can explore force without creating torque (pushing straight through a pivot), but torque always requires both force and distance from a pivot.

Do I have to use perpendicular force to calculate torque?

No, but if your force is not perpendicular, you must multiply by the sine of the angle between the force and the lever arm. Perpendicular force (90 degrees) is the most efficient because sin(90°) = 1, so you use the full force. Any other angle reduces the effective torque.

Why does a longer wrench make tightening a bolt easier?

A longer wrench increases the distance from the pivot (the bolt) to where you explore force. Since torque equals force times distance, a longer wrench produces more torque with the same push. You can tighten a bolt with less effort using a long wrench than a short one.

How do I measure the distance for the torque formula?

Measure the perpendicular distance from the pivot point to the line along which the force acts. For a wrench on a bolt, measure from the center of the bolt to where you push, straight across. For a door, measure from the hinge to where you push. The distance must be perpendicular to the direction of the force.

What units should I use for torque calculations?

Use either metric (newtons and meters, giving newton-meters) or imperial (pounds-force and feet, giving foot-pounds). Do not mix units. If you have force in newtons and distance in feet, convert one of them first so both are in the same system.