Mechanics 2: Linear and Rotational Dynamics - 3D Math Primer for Graphics and Game Development - page 609

Archimedes discovered. But what effect does changing l have on α, the
angular acceleration of the apparatus? It, too, is directly proportional: if
we push on the disk at a point twice as far out, the angular acceleration is
twice as much. This is clear from the relation α = a ⊥ /r. If this seems too
obvious to bother pointing out, then keep reading.
Let's summarize what we've found out. When a force is applied to an
object, it has a tendency to rotate that object. This “cause” of angular
acceleration is known as torque, which we denote by using the Greek letter
tau: 18 τ. Although the linear acceleration of a body as a result of an applied
force does not depend on where the force is applied, the amount of torque
that results from an applied force depends on how effectively the force is
applied. The effectiveness of the force to create rotational acceleration—the
magnitude of the torque—depends on several factors:

• It is proportional to the magnitude of the applied force f .

• It is proportional to the length of the lever arm l , which is the vector

from the fulcrum to the point of application of the force.

• Only the portion of the force perpendicular to the lever arm counts.

Equivalently, the torque is proportional to sinφ, where φ is the angle
between f and l .

In two dimensions, we can state this succinctly by

Torque in two
dimensions

τ = Fl sinφ. (12.24)

The dimensions of torque are not the same as force. Torque has units
of “force times length.” The SI unit for torque is the Newton meter.
(This is dimensionally equivalent to the joule, but torque and energy are
distinct concepts, and the joule is not really the proper unit to use for
torque.)
In might not be intuitively obvious that torque increases with the length
of the lever arm. For example, your intuition might be inclined to tell you
that it would be more di cult to push at an increased radius because you
have to push faster just to keep up. If so, your intuition is wrong, but
don't feel bad. Our experience with force is often via an everyday type of
physical push like we might make with our hands against some object. But
this is not necessarily a good example because the push must move faster
and faster as the object accelerates in order to maintain contact. But the
speed of some source of force has nothing to do with the magnitude of the
force itself. So try replacing the physical push with either a gust of wind or
a quick “thump” (an impulse). You can conduct this experiment on a door.

18 It rhymes with “cow”