Game Development Reference
In-Depth Information
10.2.1
T ORQUE
Torque (also sometimes called “moments”) can be thought of as a twisting force.
You may be familiar with a car that has a lot of torque: it can apply a great deal of
turning force to the wheels. An engine that can generate a lot of torque will be better
at accelerating the spin of the wheels. If the car has poor tires, this will leave a big black
mark on the road and a lot of smoke in the air; with appropriate grip, this rotation
will be converted into forward acceleration. In either case the torque is spinning the
wheels, and the forward motion is a secondary effect caused by the tires gripping the
road.
In fact torque is slightly different from force. We can turn a force into a torque—
that is, a straight push or pull into a turning motion. Imagine turning a stiff nut with
a wrench: you turn the nut by pushing or pulling on the handle of the wrench. When
you turn up the volume knob on a stereo, you grip it by both sides and push up with
your thumb and down with your finger (if you're right-handed). In either case you
are applying a force and getting angular motion as a result.
The angular acceleration depends on the size of the force you exert and how far
from the turning point you apply it. Take the wrench and nut example: you can undo
the nut if you exert more force onto the wrench or if you push farther along the handle
(or use a longer-handled wrench). When turning a force into a torque, the size of the
force is important, as is the distance from the axis of rotation.
The equation that links force and torque is
τ
=
p f ×
f
[10.1]
where f is the force being applied, and p f is the point at which the force is being
applied, relative to the origin of the object (i.e., its center of mass, for our purposes).
Every force that applies to an object will generate a corresponding torque. When-
ever we apply a force to a rigid body, we need to use it in the way we have so far: to
perform a linear acceleration. We will additionally need to use it to generate a torque.
If you look at equation 10.1, you may notice that any force applied so that f and p f are
in the same direction will have zero torque. Geometrically this is equivalent to saying
that if the extended line of the force passes through the center of mass, then no torque
is generated. Figure 10.1 illustrates this. We'll return to this property in section 10.3
when we combine all the forces and torques.
In three dimensions it is important to notice that a torque needs to have an axis.
We can apply a turning force about any axis we choose. So far we've considered cases
such as the volume knob or nut where the axis is fixed. For a freely rotating object,
however, the torque can act to turn the object about any axis. We give torques in a
scaled axis representation:
a d
where a is the magnitude of the torque and d is a unit-length vector in the axis around
which the torque applies. We always consider that torques act clockwise when looking
τ
=
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