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For example, according to the physical defnition of work, if one holds a hundred-
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pound weight of the ground for a while, he is doing no work. Nevertheless,
everyone knows that he begins to sweat, shake, and breathe harder, as If he were
running up a fÑight of stairs. Yet running upsfairs 7s considered as doïng work
(in running đowønstøirs, one gets work out of the world, according to physics),
but in simply holding an object in a fñxed position, no work is done. Clearly, the
physical defnition of work difers from the physiological defñnition, for reasons
we shall briely explore.
Tt is a fact that when one holds a weight he has to do “physiological” work.
'Why should he sweat? Why should he need to consume food to hold the weight
up? Why is the machinery inside him operating at full throttle, just to hold
the weight up? Actually, the weight could be held up with no efort by just
placing it on a table; then the table, quietly and calmly, without any supply of
energy, is able to maintain the same weight at the same heightl The physiological
situation is something like the following. There are two kinds of muscles in the
human body and in other animals: one kind, called strøted or skeletal muscle, 1s
the type of muscle we have in our arms, for example, which is under voluntary
control; the other kind, called srmmoo£h musele, is like the muscle in the intestines
or, in the clam, the greater adductor musecle that closes the shell. 'Phe smooth
museles work very slowly, but they can hold a “set”; that 1s to say, if the clam
tries to close its shell in a certain position, it will hold that position, even if there
is a very great force trying 0o change it. It will hold a position under load for
hours and hours without getting tired because it is very much like a table holding
up a weight, it “sets” into a certain position, and the molecules just lock there
temporarily with no work being done, no efort being generated by the clam.
The fact that we have to generate efort to hold up a weight is simply due to the
design of striated muscle. What happens is that when a nerve impulse reaches a
mmuscle fiber, the fñber gives a little twitch and then relaxes, so that when we hold
something up, enormous volleys of nerve impulses are coming in to the muscle,
large numbers oŸ twitches are maintaining the weight, while the other fñbers relax.
W© can see this, of course: when we hold a heavy weight and get tired, we begin
to shake. “The reason is that the volleys are coming irregularly, and the muscle
1s tired and not reacting fast enough. Why such an ineficient scheme? We do
not know exactly why, but evolution has not been able to develop ƒøs¿ smooth
muscle. Smooth muscle would be mụuch more efective for holding up weights
because you could just stand there and it would lock in; there would be no work
involved and no energy would be required. However, it has the disadvantage that
1t 1s very slow-operating.
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Returning now to physics, we may ask +0 we want 6o calculate the work
done. The answer is that it is interesting and useful to do so, since the work done
on a particle by the resultant of all the forces acting on it is exactly equal to the
change in kinetic energy of that particle. That is, iŸ an object is being pushed, it
picks up speed, and
A(0?)= ¬ .As.
14-2 Constrained motion
Another interesting feature of forces and work is this: suppose that we have
a sloping or a curved track, and a particle that must move along the track, but
without friction. Ôr we may have a pendulum with a string and a weight; the
string constrains the weight to move in a circle about the pivot point. “The pivot
point may be changed by having the string hit a peg, so that the path of the
weight is along two circles of diferent radii. Thhese are examples of what we call
liacd, [riclionless constraints.
In motion with a ñxed frictionless constraint, no work is done by the constraint
because the forces of constraint are always at right angles to the motion. By
the “forces of constraint” we mean those forces which are applied to the object
directly by the constraint itself—the contact force with the track, or the tension
in the string.
'The forces involved in the motion of a particle on a slope moving under the
inÑuence of gravity are quite complicated, since there is a constraint Íorce, a
gravitational force, and so on. However, if we base our calculation of the motion
on conservation of energy and the grauftational ƒorce alone, we get the right result.
This seems rather strange, because it is not strictly the right way to do it—we
should use the resulfamt force. Nevertheless, the work done by the gravitational
force alone will turn out to be the change in the kinetic energy, because the work
done by the constraint part of the force is zero (Eig. 14-1).
FORCE OF `
CONSTRAINT FORCE OE
GRAVITY
Fig. 14-1. Forces acting on a sliding body (no friction).
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The important feature here is that if a force can be analyzed as the sum of
two or more “pieces” then the work done by the resultant force in going along a
certain curve is the sum of the works done by the various “component” forces into
which the force is analyzed. 'Thus if we analyze the force as being the vector sum
of several efects, gravitational plus constraint forces, etc., or the ø-component of
all forces and the -component of all forces, or any other way that we wish to
split it up, then the work done by the net force is equal to the sum of the works
done by all the parts into which we have divided the force in making the analysis.
14-3 Conservative Íorces
In nature there are certain forces, that of gravity, for example, which have a
very remarkable property which we call “eonservative” (no political ideas involved,
1E is again one oŸ those “crazy words”). IÝ we calculate how much work is done
by a force in moving an object from one poiïnt to another along some curved
path, in general the work depends upon the curve, but in special cases it does
not. lf it does not depend upon the curve, we say that the force is a conservative
force. In other words, if the integral of the force times the distance in goỉng from
position 1 to position 2 in Eig. 14-2 is calculated along curve 4 and then along Ö,
we get the same number of Joules, and if this is true for this pair of points on
cucrU curue, and 1f the same propositlon works no matter thích pa#r öŸ poin‡s
we use, then we say the fÍorce is conservative. In such circumstances, the work
integral going from 1 to 2 can be evaluated in a simpÌe manner, and we can give
a formula for the result. Ordinarily it is not this easy, because we also have to
specify the curve, but when we have a case where the work does not depend on
the curve, then, of course, the work depends only upon the pos/fzons of 1 and 2.
P C Z—>ce 2
Fig. 14-2. Possible paths between two points ¡in a field of force.
To demonstrate this idea, consider the following. We take a “standard”
point Ð, at an arbitrary location (Fig. 14-2). Then, the work line-integral rom 1