Torque is the rotational effect of a force. It describes how strongly a force tends to turn an object around a pivot, axle, or axis of rotation. When you push a door open, loosen a bolt with a wrench, or pedal a bicycle, you apply torque.
In physics, torque helps explain why some forces produce rotation more effectively than others. The result depends not only on how much force you apply but also on where you apply it and the direction in which you push or pull.
Torque is a fundamental concept in mechanics, the branch of physics that studies how objects move and respond to forces. It is essential for understanding everything from simple hand tools to engines, electric motors, and rotating machinery.
What is torque in physics?
Torque measures a force’s tendency to rotate an object about a specified point or axis. It is the rotational counterpart of force in linear motion, although the two quantities describe different effects.
A force can make an object accelerate in a straight line, change its direction, or cause it to rotate. Whether a force produces rotation depends on its position relative to the axis and the direction in which it acts.
Consider a door attached to hinges. If you push near the hinges, the door is difficult to turn. If you apply the same force near the handle, the door opens more easily. The force is identical in both cases, but the second push produces more torque because it acts farther from the axis of rotation.
Torque can also exist without producing movement. For example, pushing hard on a locked door creates torque about its hinges, but the door remains stationary because the resisting forces prevent it from rotating. Torque describes the turning effect of a force, not necessarily the resulting motion.
The torque formula
The magnitude of torque is calculated by multiplying the applied force by the perpendicular distance from the axis of rotation to the force’s line of action.
The formula is:
τ = rF sin θ
Here:
- τ (tau) represents torque.
- r is the distance from the axis of rotation to the point where the force is applied.
- F is the magnitude of the applied force.
- θ is the angle between the position vector and the force.
The position vector points from the axis of rotation toward the point where the force is applied. The angle θ therefore describes the direction of the force relative to that position.
An equivalent form uses the perpendicular lever arm, d:
τ = Fd
In this expression, d is the shortest perpendicular distance between the axis of rotation and the force’s line of action. This form is often the easiest to use in practical situations.
For example, suppose you apply a 20-newton force perpendicular to a wrench that is 0.30 meters long. The torque about the bolt is:
τ = Fd
τ = 20 N × 0.30 m
τ = 6 N·m
The applied force produces a torque with a magnitude of 6 newton-meters.
This calculation assumes that the entire 0.30-meter length is the perpendicular distance from the bolt’s axis to the force’s line of action. If you push at an angle, the perpendicular lever arm becomes smaller, reducing the torque for the same force and distance from the axis.
What determines how much torque a force produces?
Three factors determine the magnitude of torque: the strength of the force, the distance from the axis of rotation, and the angle at which the force acts.
Force strength: Increasing the force increases torque when the other conditions remain unchanged. Pushing harder on a wrench generally produces a greater turning effect, provided the force is applied in the same direction and at the same location.
Distance from the axis: Applying a force farther from the axis increases torque when the force’s direction remains unchanged. This is why long-handled tools can make stubborn bolts easier to loosen. A longer handle provides a larger lever arm, allowing the same force to create more torque.
Angle of application: A force produces its greatest torque when it acts perpendicular to the position vector. In the equation τ = rF sin θ, the sine of the angle reaches its maximum value of 1 at 90 degrees.
When the force acts at 90 degrees, the formula simplifies to τ = rF. When it acts parallel or antiparallel to the position vector, the angle is 0 or 180 degrees, and the torque about that point is zero. The force’s line of action passes through the axis, so it has no lever arm.
These principles explain why pushing sideways on a door near its handle is effective, while pushing directly toward the hinges is not. The direction and location of the force matter just as much as its magnitude.
What are the units of torque?
The standard SI unit of torque is the newton-meter, written N·m. One newton-meter is the torque produced by a one-newton force acting with a perpendicular lever arm of one meter.
Because force is measured in newtons and distance in meters, the torque formula gives the unit:
N·m = N × m
Torque has the same dimensions as energy, which is measured in joules. However, torque and energy are different physical quantities. Torque describes a turning effect, whereas energy describes the capacity to do work or produce change.
For this reason, torque is conventionally expressed in newton-meters rather than joules, even though 1 N·m has the same base-unit dimensions as 1 joule.
In the United States, you may also encounter foot-pounds (ft·lb) when reading specifications for engines, automotive fasteners, or power tools. One foot-pound of torque is the turning effect of a one-pound-force applied at a perpendicular distance of one foot.
It is important to distinguish foot-pounds of torque from other quantities expressed using the same combination of customary units. The context determines whether a value describes rotational torque or work.
Torque and the direction of rotation
Torque has both magnitude and direction. Depending on how a force is applied, it can tend to rotate an object clockwise or counterclockwise when viewed from a particular side.
For rotation in a flat plane, physicists often assign one sign to counterclockwise torque and the opposite sign to clockwise torque. A common convention treats counterclockwise torque as positive and clockwise torque as negative, although the choice of signs is a matter of convention.
For more general three-dimensional motion, torque is represented as a vector:
τ⃗ = r⃗ × F⃗
The cross product, represented by ×, produces a vector perpendicular to the plane containing the position and force vectors. Its direction follows the right-hand rule: curl the fingers of your right hand from the position vector toward the force vector through the smaller angle, and your thumb indicates the torque direction.
This vector description is useful for rotating shafts, wheels, robotic joints, and other systems in which rotation can occur about different axes.
When several forces act on an object, each can contribute torque. The total, or net, torque is the vector sum of the individual torques about the chosen reference point or axis.
If clockwise and counterclockwise torques oppose each other, they can partially or completely cancel. An object in rotational equilibrium has zero net torque, meaning there is no angular acceleration. If the object is initially at rest, it remains at rest rotationally, assuming the net torque stays zero.
Everyday examples of torque
Torque appears in ordinary activities whenever a force creates or opposes rotation. The same physical principles apply whether the object is a small hand tool or a large machine.
Opening a door: A door rotates around its hinges. Pushing near the handle creates more torque than pushing near the hinges because the handle is farther from the rotational axis. Pushing perpendicular to the door’s surface also tends to maximize the turning effect for a given force at that location.
Using a wrench: A wrench turns a nut or bolt around its axis. A longer wrench provides a greater lever arm, so less force is needed to produce a given torque. This is why mechanics sometimes use longer handles to loosen tight fasteners. The tool and fastener must still be strong enough to withstand the applied force without bending, slipping, or breaking.
Riding a bicycle: When a cyclist pushes down on a pedal, the force creates torque about the crank axle. The torque depends on the effective distance from the axle to the pedal and the direction of the force. The crank converts the rider’s force into rotational motion that helps drive the bicycle’s drivetrain.
Turning a steering wheel: A force applied near the outer edge of a steering wheel produces more torque than the same force applied closer to its center. The wheel’s radius provides the lever arm. This arrangement allows a driver to turn the wheel without having to apply the force directly at the steering column.
Using a screwdriver: A screwdriver applies torque to a screw around the screw’s axis. A larger handle can make it easier to apply a turning force because the hand can act at a greater radius. The amount of torque transferred also depends on grip, friction, and whether the screwdriver tip fits the screw head properly.
Balancing on a seesaw: A seesaw rotates around a central pivot. A person sitting farther from the pivot creates more torque than a person of the same weight sitting closer to it. Two people can balance each other when their opposing torques are equal, even if they have different weights, provided their distances from the pivot compensate appropriately.
Opening a jar: Twisting a jar lid creates torque around the lid’s central axis. A firm grip helps transfer force to the lid, while a wider lid or gripping surface can provide a larger effective radius. Friction between the hand and lid is essential; without enough friction, the hand may slip instead of turning the lid.
Torque and rotational motion
Torque is closely related to angular acceleration, which describes how quickly an object’s rotational speed changes. In linear motion, Newton’s second law relates net force to mass and acceleration. For rotation about a fixed axis, the corresponding relationship is:
τₙₑₜ = Iα
Here:
- τₙₑₜ is the net torque about the axis.
- I is the moment of inertia about that axis.
- α (alpha) is the angular acceleration.
The moment of inertia describes how difficult it is to change an object’s rotational motion. It depends on the object’s mass and how that mass is distributed relative to the axis of rotation.
For example, a mass concentrated far from an axis generally produces a larger moment of inertia than the same mass concentrated near the axis. As a result, the same net torque produces less angular acceleration when the moment of inertia is larger.
This principle helps explain why a heavy flywheel can resist changes in rotational speed and why the distribution of mass matters when spinning wheels, rotating machinery, and athletic equipment.
The relationship τₙₑₜ = Iα applies directly to rotation about a fixed axis when the moment of inertia is constant. More complex rotational motion, particularly when an object changes its orientation in three dimensions, requires a fuller treatment of rotational dynamics.
Torque does not determine rotational speed by itself. Instead, net torque influences how rotational speed changes. An object can rotate rapidly with zero net torque if it is already moving at a constant angular velocity and no net torque acts to change that motion.
Torque, work, and power
Torque, work, and power are related concepts, but they describe different aspects of motion.
Torque describes a force’s tendency to cause rotation. Rotational work describes energy transferred when a torque acts through an angular displacement. For a constant torque acting along a fixed axis, the work done is:
W = τθ
Here, W is work, τ is the torque component along the axis of rotation, and θ is the angular displacement measured in radians.
For example, if a constant torque of 4 N·m turns a shaft through an angle of 2 radians, the work done by that torque is 8 joules.
Power measures how quickly work is done or energy is transferred. For rotation about a fixed axis, instantaneous mechanical power is:
P = τω
Here, P is power and ω (omega) is angular velocity, measured in radians per second. This relationship shows that a rotating machine can deliver more power by producing more torque, rotating faster, or both, assuming the relevant quantities refer to the same axis and the torque acts in the direction of rotation.
This distinction is particularly useful when comparing engines and electric motors. Torque indicates their turning effect, while power describes the rate at which they can transfer mechanical energy.
A motor that produces substantial torque at a low rotational speed may have less power output than one producing the same torque at a higher speed. Likewise, a machine can produce high power through a combination of moderate torque and high angular velocity.
Torque in engines, motors, and machines
Torque is a key specification for engines and electric motors because it describes their ability to exert a turning effect on a shaft. In a vehicle, engine torque is transmitted through the drivetrain to the wheels, where it contributes to the force available to accelerate the vehicle.
Gearing can change the torque and rotational speed delivered to the wheels. A gear ratio that reduces rotational speed can increase output torque, apart from losses caused by friction and other inefficiencies. The trade-off is that the output shaft turns more slowly than the input shaft.
This is why lower gears are useful when starting a vehicle, climbing a steep hill, or moving a heavy load. They help deliver greater wheel torque for a given engine torque, although the actual result also depends on engine speed, gear selection, traction, and drivetrain design.
Electric motors also produce torque, which can turn fans, pumps, washing machines, power tools, and industrial equipment. The torque available from a motor varies with its design and operating conditions. Some motors are designed to deliver strong starting torque, while others are optimized for particular speed ranges or applications.
In all these systems, engineers consider torque alongside speed, power, efficiency, heat, and mechanical strength. Excessive torque can damage shafts, gears, fasteners, or other components, so machines are designed with appropriate operating limits.
Torque and rotational equilibrium
An object is in rotational equilibrium when the net torque acting on it is zero. In this condition, its angular acceleration is zero, although the object may be stationary or rotating at a constant angular velocity.
A balanced seesaw provides a familiar example. If one person produces a clockwise torque of 150 N·m and another produces an equal counterclockwise torque of 150 N·m about the same pivot, the net torque is zero.
If the opposing torques are unequal, the seesaw experiences a net torque and tends to develop angular acceleration. Its actual motion also depends on its moment of inertia and any other forces or constraints acting on the system.
For a rigid object to remain completely at rest in mechanical equilibrium, zero net torque alone is not sufficient. The net force must also be zero. An object can have zero net torque while still accelerating in a straight line if an unbalanced force acts on it.
The choice of reference point matters when calculating torque. For a rigid body in static equilibrium, the net torque is zero about any reference point. In other situations, torques calculated about different points can differ because the object may also experience a nonzero net force.
Common misconceptions about torque
One common misconception is that torque is simply force multiplied by distance. The formula τ = rF applies only when the force is perpendicular to the position vector. In the general case, the correct magnitude is τ = rF sin θ, or equivalently τ = Fd using the perpendicular lever arm.
Another misconception is that greater torque always means greater rotational speed. Torque affects angular acceleration, not rotational speed directly. Rotational speed also depends on the object’s existing motion, its moment of inertia, and the torques acting on it over time.
It is also incorrect to treat torque and energy as interchangeable merely because their SI units have the same dimensions. Torque describes a rotational effect, while energy describes the ability to do work or the amount of work transferred. The distinction matters in equations, measurements, and engineering specifications.
Finally, torque does not always cause an object to rotate. A force can create torque while a resisting force or opposing torque keeps the object stationary. A tight bolt, for example, can experience substantial applied torque without turning if the opposing effects are sufficient to prevent motion.
Why torque matters in physics
Torque provides a practical way to understand how forces produce rotation. It explains why a longer wrench makes turning easier, why a seesaw balances when opposing torques match, and why motors and engines are rated partly by their turning capability.
The central principle is straightforward: the rotational effect of a force depends on its magnitude, its direction, and its perpendicular distance from the axis of rotation. The formula τ = rF sin θ captures these factors in a single relationship.
By connecting force to angular acceleration, torque also helps explain how rotating systems behave, how machines transfer energy, and how engineers design components to move loads safely and efficiently. It is one of the essential ideas linking everyday mechanical experiences to the broader laws of physics.
