Torque and Lever Arm Distance
I know that torque is produced when a force is applied a
certain distance from the pivot point of the lever arm (unless it is
applied at 0 degrees or 180 degrees). I also know that the further
away from the fulcrum the force is applied then the greater the
torque resulting. My question is: WHY? Why does the distance the
force is applied from the fulcrum heighten the torque? What causes
this to be the case?
The effect can be explained with either acceleration or work. I prefer
the second because it does not require worrying about NET force. It can
be applied to a single force.
A force is applied at a certain distance from the pivot. The force
pushes on a portion of the object that moves along the tangent of a
circle, so the component directed toward the pivot does no work. Work
done, over a very short distance, is tangential force times distance
along the circumference. Tangential force is force perpendicular to the
radius. Distance is radius times a small angle of rotation (in
radians). For truly rotational motion, the radius is constant. For
truly rotational motion, work equals tangential force times radius times
angle of rotation. This can then be expressed as (perpendicular force)
times angle change, or torque times angle change.
Perpendicular force is force magnitude times a sine function: torque
equals force times radius times a sine function. If you wish to use
lever arm, radius at which the force makes contact times the same sine
function yields this quantity called lever arm (portion of radius
perpendicular to the force). This allows you to write torque times
Dr. Ken Mellendorf
Illinois Central College
The simple answer is that Torque is a derived unit. It is defined based on
the other two (force and distance). But that kind of avoids the question.
A torsional system consists of a force applied at some lever arm that is
reacted by another force at another lever arm. If you "unwrap" a torsional
system, you will have a lever. Exactly the same principle is acting there.
Now let us consider the "work" done at each end. Work is force times
distance traveled. If I have a lever with a 2:1 length ratio, and I push
down with a force of 100 lb., I can lift 200 lb. on the other end. But look
at the work done: I have to push the 100 lb. end down twice as far as the
200 lb. end comes up. the work done on one end is 100 lb * 2 feet at one
end, and 200 lb. * 1 foot on the other. The work done on either side is
equal at 200 foot-pounds. Work and energy are different terms for one
another, and can be equated (search for "energy" in Wikipedia, and scroll
down to the section on forms of energy and transformation of energy for the
theory behind what engineers call the "work-energy equation"), so I have not
broken any of the laws of thermodynamics by having more force coming out
than went in. It is just that something else must be less, such that the
energy or work on each end is the same (less losses, of course). Of course,
I have not included any losses, but this simplistic picture gives you the idea.
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Update: June 2012