Torque
''This article is about the
physical concept. For another meaning see
Torque (jewellery)
----
The concept of
torque in
physics, also called
moment or
couple, originated with the work of
Archimedes on
levers. Informally, torque can be thought of as "rotational force". The rotational analogues of
force,
mass and
acceleration are torque,
moment of inertia and
angular acceleration. The
force applied to a lever, multiplied by its distance from the lever's fulcrum, is the torque. For example, a force of three
newtons applied two
metres from the fulcrum exerts the same torque as one newton applied six metres from the fulcrum. This assumes the force is in a direction at right angles to the straight lever. More generally, one may define torque as the
cross product:
:
where
r is the
vector from the
axis of rotation to the point on which the force is acting
F is the vector of
force.
Units
Torque has dimensions of
distance ×
force; the same as
energy. However, the
SI units of torque are usually stated as "
newton-
metres" rather than
joules. Of course this is not simply a coincidence. A torque of 1 N·m applied through a full revolution will require an
energy of exactly 2π joules. Mathematically,
:
where
E is the energy
θ is the angle moved, in
radians.
Other non-SI units of torque include "
pound-force-
feet"
A very useful special case, often given as the definition of torque in fields other than physics, is as follows:
:
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. Note that 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:
:
Image:moment arm.png
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 in the direction best suited to turning bolts.
If the force is at an angle θ from the perpendicular then, from the definition of cross product, the magnitude of the torque arising is:
:
For an object to be at
static equilibrium, not only must the sum of the forces be zero, but also the sum of the torques (moments). 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 is the time-
derivative of
angular momentum, just as force is the time derivative of
linear momentum. For multiple torques acting simultaneously:
:
where
L is angular momentum.
Torque on a rigid body can be written in terms of its
moment of inertia and its
angular velocity :
:
so if
is constant,
:
where
α is
angular acceleration, a quantity usually measured in
rad/
s2.
Machine torque
Torque is important in the design of machines such as
engines. The measurement of torque is also important in automotive engineering, being concerned with the
transmission of
power from the engine through the drive train to the wheels of a vehicle. Torque (and power output) can be measured with a
dynamometer.
A
torque wrench is used where the tightness of
screws and bolts is crucial. Torque is also the easiest way to explain
mechanical advantage in just about every
simple machine except the
pulley.
See also
da:Drejningsmoment
de:Drehmoment
es:Par de giro
fr:Moment (mécanique)
ja:トルク
nl:Koppel (natuurkunde)
pl:Moment siły
sl:Navor
sv:Vridmoment
zh-cn:力矩
Category:Mechanical engineering
Category:Physical quantity