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Re: [Phys-l] 'Torque' question



John has, I think, answered the basic question, but since the Indiana
Academy is a High School, let me simplify a little more.

John's dE = FRd(theta) can be viewed as the rotational equivalent of work --
Torque x rotational displacement. Normally one needs to integrate Torque x
d(theta) just like one needs to integrate Force x ds, but with the
conditions of constant force applied always perpendicularly, we have Work =
FR(theta). [Don't get confused here. You learned that Torque is RxF =
RFsin(theta), but there theta is the angle between R and F. In John's and
my answer, theta is the angle through which the object (door) swings--the
angular displacement. ] R(theta) is the arc length of the rotational swing.
Theta will be the same (by assumption) no matter where the force is
applied. Therefore, for small R the arc length is small so to get a given
amount of work requires a larger force than if R is large with a longer arc
length. The ultimate answer remains conservation of energy.

Rick (who also first wondered if the question was meta-physical, but
decided it was the question answered by JD and above.)

--------------------------------------------------
From: "John Denker" <jsd@av8n.com>
Sent: Monday, July 12, 2010 2:46 AM
To: "Forum for Physics Educators" <phys-l@carnot.physics.buffalo.edu>
Subject: Re: [Phys-l] 'Torque' question

On 07/11/2010 06:33 PM, Fakhruddin, Hasan wrote:

Say a force is applied on an object to rotate it about a hinge. Why
does it get easier (like needing less force) to rotate the object as
the force is applied farther from the hinge?

I'm curious as to how the question came up.

The key word there is "why". In all questions of this
sort, I assume the word "why" is an idiomatic expression,
and the intended meaning could be more explicitly expressed
as:
-- How do we know (that the force will be less).
-- How do we predict (that the force will be less).
-- How do we calculate (that the force will be less).

There are lots of good ways of knowing that. Perhaps the
simplest is PVW (the principle of virtual work) which is
a corollary of the conservation of energy.

Under mild assumptions we can differentiate the energy
∂E ∂E
dE = ------ dx + ------ dS
∂x | S ∂S | x

and the first term on the RHS _defines_ what we mean
by force:

dE = - F dx + T dS

where

F := - ∂E/∂x|S

Meanwhile, geometry tells us that for rotational motion
we have
dx = R dθ
so
dE = F R dθ
where for simplicity we assume F is perpendicular to R.
We also assume dS=0, since this is a simple mechanics
problem, not a thermodynamics problem.

By construction dθ is the same everywhere (rigid lever)
and by conservation of energy dE at one place is the
same as dE at another, so
F1 R1 = F2 R2
which suffices to answer the original question.

=========

If you really meant to ask "why" in the sense of cause
and effect, the question is not a physics question. It
has no answer within physics, and probably no answer at
all.

As Galileo pointed out in 1638: Physics is obliged to
say what happens. It might or might not say how it
happens, and it almost never says why. This is what
sets physics apart from philosophy and metaphysics.
Newton famously summarized this by saying "hypotheses
non fingo".

The laws of motion do not express cause and effect.
http://www.av8n.com/physics/causation.htm
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