Moment of Force

Torque, also known as the moment of force, is a measure of the rotational force of an object. It is the product of the magnitude of the force applied and the distance from the point of application to the point around which the force is being measured. This concept was first studied by Archimedes, who famously said, "Give me a lever and a place to stand and I will move the Earth." In three dimensions, torque is a pseudovector represented by the Greek letter tau ({\displaystyle {\boldsymbol {\tau }}}{\boldsymbol {\tau }}). It is the cross product of the position vector and the force vector and is affected by the force applied, the lever arm connecting the two points, and the angle between the force and lever arm. Torque is an important concept in physics and mechanics, as it helps to understand the rotational motion and the conservation of energy of an object.

The term "torque" comes from the Latin word "torquēre," which means "to twist." It was first introduced by James Thomson in 1884 and used in Silvanus P. Thompson's Dynamo-Electric Machinery the same year. Torque is defined as the force that produces or tends to produce a twisting motion around an axis. It is often referred to as "moment of force," or simply "moment," in the fields of mechanical engineering in the UK and US. The concept of torque is used to describe the action of a single, definite twist applied to turn a shaft, rather than the more complex idea of applying a linear force or pair of forces with leverage. In the US, the term "torque" is commonly used in the field of physics. In contrast, the term "moment" is used in French, as attested by Siméon Denis Poisson's Traité de mécanique in 1811.

Torque is a measure of the rotational force applied to an object, such as a lever. It is calculated by multiplying the force applied perpendicularly to the lever by the distance from the fulcrum, or pivot point, of the lever (also known as the lever arm). For example, a force of three newtons applied two meters from the fulcrum exerts the same torque as a force of one newton applied six meters from the fulcrum. To determine the direction of the torque, the right hand grip rule can be used. To use this rule, curl the fingers of your right hand from the direction of the lever arm towards the direction of the applied force. The thumb will point in the direction of the torque.

Torque is typically measured in units of force times distance, such as newton-meters (N⋅m) or pound-feet (lbf-ft). The International System of Units (SI) recommends using newton-meters, but the traditional imperial and US customary units for torque are pound-feet and pound-inches (lbf-in). In the US, torque is often referred to as foot-pounds (lb-ft or ft-lb) or inch-pounds (in-lb). It is important to pay attention to context and the hyphen in the abbreviation to distinguish torque from other quantities such as energy or moment of mass.

In fields outside of physics, torque is often defined as the product of the moment arm and the force. The moment arm is the distance between the point where the force is applied and the center of rotation, as shown in the figure. This definition only gives the magnitude of the torque, not its direction, which can make it difficult to use in three-dimensional cases. When the force is applied perpendicularly to the displacement vector (the distance between the initial and final positions), the moment arm is equal to the distance to the center of rotation, and the torque is at its maximum for the given force. The equation for the magnitude of torque in this case is: torque = (distance to center) x (force). For example, if a person applies a force of 10 N at the end of a wrench that is 0.5 m long (or a force of 10 N acting 0.5 m from the point of rotation on a wrench of any length), the torque will be 5 N⋅m. This assumes that the person moves the wrench by applying force in the plane of movement and perpendicular to the wrench.

In order for an object to be in static equilibrium, the sum of all forces acting on it must be zero, and the sum of all torques (moments) about any point must also be zero. This means that in a two-dimensional situation with horizontal and vertical forces, there must be three equations to solve the problem: one for the sum of the horizontal forces (ΣH = 0), one for the sum of the vertical forces (ΣV = 0), and one for the sum of the torques (Στ = 0). When the net force on a system is zero, the torque measured from any point in space will be the same. For example, the torque on a loop of current in a uniform magnetic field is the same regardless of the reference point used.

Torque is a key specification for engines, as it determines the power output of the engine. It is calculated by multiplying the torque by the angular speed of the drive shaft. Internal combustion engines typically produce useful torque within a limited range of rotational speeds (usually from around 1,000-6,000 rpm for a small car). The torque output of an engine can be measured over this range using a dynamometer, and the resulting curve is known as the torque curve. In contrast, steam engines and electric motors tend to produce maximum torque at low rpm, with the torque decreasing as the rotational speed increases due to factors such as increasing friction. These types of engines are able to start heavy loads from a standstill without the need for a clutch.

Torque can be used to do work when it is allowed to act through an angular displacement, just as a force can do work when it is allowed to act through a distance. The relationship between torque, angular speed, and power can be expressed algebraically, and this relationship can be observed in practical examples such as bicycles. In a bicycle, the rider provides input power by turning the pedals, which cranks the front sprocket (chainring). This input power is equal to the product of the angular speed (the number of pedal revolutions per minute times 2π) and the torque at the spindle of the crankset. The drivetrain of the bicycle transmits this input power to the road wheels, which convey the received power to the road as the output power of the bicycle. Depending on the gear ratio of the bicycle, the input (torque, angular speed) pair is converted to an output (torque, angular speed) pair. For example, using a larger rear gear or switching to a lower gear on a multi-speed bicycle will decrease the angular speed of the road wheels while increasing the torque, resulting in no change in the power output. In the International System of Units (SI), the unit of power is the watt, the unit of torque is the newton-meter, and the unit of angular speed is the radian per second. It is important to note that the newton-meter is dimensionally equivalent to the joule, the unit of energy, but it is assigned to a vector (torque) rather than a scalar (energy). This distinction is addressed in orientational analysis, which treats the radian as a base unit rather than a dimensionless unit.

It may be necessary to use a conversion factor when working with different units of power or torque. For example, if rotational speed (revolutions per time) is used instead of angular speed (radians per time), you must multiply by a factor of 2π radians per revolution. In the following formulas, P represents power, τ represents torque, and ν (nu) represents rotational speed. Some people, such as American automotive engineers, use horsepower for power, foot-pounds for torque, and rpm for rotational speed. The conversion factor (in foot-pounds per minute) changes depending on the definition of horsepower; for example, using metric horsepower, it becomes approximately 32,550. If you are using other units, such as BTU per hour for power, you will need a different custom conversion factor.

For a rotating object, the linear distance covered at the circumference of rotation is equal to the product of the radius and the angle covered. This can be expressed as: linear distance = radius × angular distance. It can also be expressed as: linear distance = linear speed × time = radius × angular speed × time. Using the definition of torque (torque = radius × force), we can rearrange this equation to determine force = torque ÷ radius. In this equation, the radius (r) and time (t) have been eliminated. However, it is important to note that the angular speed must be in radians per unit of time, based on the assumption of a direct relationship between linear speed and angular speed at the beginning of the derivation. If the rotational speed is measured in revolutions per unit of time, the linear speed and distance are increased proportionately by 2π in the above derivation, resulting in the following equation:

If torque is in newton meters and rotational speed is in revolutions per second, this equation gives power in newton meters per second, or watts. If Imperial units are used, and if torque is in pound-feet and rotational speed is in revolutions per minute, the equation gives power in foot-pounds per minute. The horsepower form of the equation can be derived by applying the conversion factor 33,000 ft⋅lbf/min per horsepower:

The principle of moments, also known as Varignon's theorem, states that the total torque resulting from multiple forces applied to a specific point is equal to the sum of the individual torques. This can be expressed as:

τ = r1 × F1 + r2 × F2 + … + rN × FN

Based on this principle, the torques resulting from two forces acting around a pivot on an object are balanced when:

r1 × F1 + r2 × F2 = 0

There are three ways to multiply torque: by locating the fulcrum in a way that increases the length of a lever; by using a longer lever; or by using a gearset or gearbox that reduces speed and increases torque. Such a mechanism multiplies torque by reducing the rotation rate.