# Torque and angular acceleration

##### Torque
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Definitions

In w:physics, torque (τ) is also called moment), and is a vector that measures the tendency of a force to rotate an object about some axis [1] (center). The magnitude of a torque is defined as force times the length of the w:lever arm [2] (radius). Just as a force is a push or a pull, a torque can be thought of as a twist.

• The force applied to a lever, multiplied by its distance from the lever's fulcrum, is its torque. [3]
${\displaystyle {\boldsymbol {\tau }}=\mathbf {r} \times \mathbf {F} }$

where r is the particle's w:position vector relative to the fulcrum, and F is the force acting on the particles.

Units

As with any concept defined by a formula, the units Torque (force times distance ) can be determined by the formula (e.g., newton meter inSI units)[4] Even though the order of "newton" and "meter" are mathematically interchangeable, the BIPM (Bureau International des Poids et Mesures) specifies that the order should be N m not m N. N·m is also acceptable.[5]

The joule, which is the SI unit for energy or work, is also defined as 1 N m, but this unit is not used for torque. While both torque and energy have the same units, one is a scalar and the other is a vector (technically (pseudo) vector)

Relation between torque and energy
${\displaystyle E=\tau \theta \ }$

where

E is the energy
τ is torque
θ is the angle moved, in radians.

Other non-SI units of torque include "pound-force-feet" or "foot-pounds-force" or "ounce-force-inches" or "meter-kilograms-force".

Moment arm
Moment arm diagram

A very useful special case, often given as the definition of torque in fields other than physics, is as follows:

${\displaystyle \tau =({\textrm {moment\ arm}})\cdot {\textrm {force}}}$

The construction of the "moment arm" is shown in the figure below, along with the vectors r and F mentioned above. The problem with this definition is that it does not give the direction of the torque but only the magnitude, and hence it is difficult to use in three-dimensional cases. If the force is perpendicular to the displacement vector r, the moment arm will be equal to the distance to the centre, and torque will be a maximum for the given force. The equation for the magnitude of a torque arising from a perpendicular force:

${\displaystyle \tau =({\textrm {distance\ to\ center}})\cdot {\textrm {force}}}$

For example, if a person places a force of 10 N on a spanner which is 0.5 m long, the torque will be 5 N m, assuming that the person pulls the spanner by applying force perpendicular to the spanner.

Force at an angle

If a force of magnitude F is at an angle θ from the displacement arm of length r (and within the plane perpendicular to the rotation axis), then from the definition of cross product, the magnitude of the torque arising is:

${\displaystyle \tau =rF\sin \theta }$

Static equilibrium

For an object to be in static equilibrium, not only must the sum of the forces be zero, but also the sum of the torques (moments) about any point. For a two-dimensional situation with horizontal and vertical forces, the sum of the forces requirement is two equations: ΣH = 0 and ΣV = 0, and the torque a third equation: Στ = 0. That is, to solve statically determinate equilibrium problems in two-dimensions, we use three equations.

Torque and power

If a force is allowed to act through a distance, it is doing mechanical work. Similarly, if torque is allowed to act through a rotational distance, it is doing work. Power is the work per unit time. However, time and rotational distance are related by the angular speed where each revolution results in the circumference of the circle being travelled by the force that is generating the torque. The power injected by the applied torque may be calculated as:

${\displaystyle {\mbox{Power}}={\mbox{torque}}\cdot {\mbox{angular speed}}\,}$

##### Angular Acceleration
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Definition

Angular acceleration is the rate of change of angular velocity over time. In SI units, it is measured in radians per second squared (rad/s2), and is usually denoted by the Greek letter alpha (${\displaystyle {\alpha }\,}$). It can be defined as either:

${\displaystyle {\alpha }={\frac {d{\omega }}{dt}}={\frac {d^{2}{\theta }}{dt^{2}}}}$ , or
${\displaystyle {\alpha }={\frac {\mathbf {a} _{T}}{r}}}$ ,

where ${\displaystyle {\omega }}$ is the angular velocity, ${\displaystyle \mathbf {a} _{T}}$ is the linear tangential acceleration, and r is the radius of curvature.

Equations of motion

For rotational motion, Newton's second law can be adapted to describe the relation between torque and angular acceleration:

${\displaystyle {\tau }=I\ {\alpha }}$ ,

where ${\displaystyle {\tau }}$ is the total torque exerted on the body, and ${\displaystyle I}$ is the mass moment of inertia of the body.

Constant acceleration:

For all constant values of the torque, ${\displaystyle {\tau }}$, of an object, the angular acceleration will also be constant. For this special case of constant angular acceleration, the below equation will produce a definitive, single value for the angular acceleration:

${\displaystyle {\alpha }={\frac {\tau }{I}}}$ .

Non-constant acceleration:

For any non-constant torque, the angular acceleration of an object will change with time. The equation becomes a differential equation instead of a singular value. This differential equation is known as the equation of motion of the system and can completely describe the motion of the object.

If the torques on an object cancel out, the net torque is zero and the angular acceleration is also zero. For example, a beam that can rotate about its axis has two forces exerted on it and therefore two torques (see figure 3 below).

1. Serway, R. A. and Jewett, Jr. J. W. (2003). Physics for Scientists and Engineers. 6th Ed. Brooks Cole. ISBN 0-53440-842-7.
2. Tipler, Paul (2004). Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics (5th ed.). W. H. Freeman. ISBN 0-7167-0809-4.
3. For example, a force of three w:newtons applied two w:meters from the fulcrum exerts the same torque as one newton applied six meters from the fulcrum. This assumes the force is in a direction at w:right angles to the straight lever. The direction of the torque can be determined by using the right hand rule: Using your right hand, curl your fingers in the direction of rotation, and stick your thumb out so it is aligned with the axis of rotation. Your thumb points in the direction of the torque vector. http://hyperphysics.phy-astr.gsu.edu/hbase/tord.html
4. "SI brochure Ed. 8, Section 5.1". Bureau International des Poids et Mesures. 2006. Retrieved 2007-04-01.
5. "SI brochure Ed. 8, Section 2.2.2". Bureau International des Poids et Mesures. 2006. Retrieved 2007-04-01.