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All work and no play...
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Up to now we have been using Newton's laws of motion to
predict the future of dynamical systems. Sir Isaac had some other
ideas about dynamics that have proven to be useful. In this
section of the program we will explore a few of them.
Let's begin with the notion of
"work". Work is defined as force multiplied by
displacement. Remember in the general case, force and
displacement are vectors. Work is a scalar quantity so the
multiplication operation we are talking about here is the dot
product as discussed back in
Vector Arithmetic. If the fellow in the picture at the right was
actually holding up his end of that bit of pipe while the machine held up the other, neither
of them would be doing any work since the force (upward) is perpendicular to the motion
(horizontal).
Consider a free particle of mass, m, at rest in
our reference frame. Next suppose that we apply a force in the x
direction to the particle and that the magnitude of that force is
a function of x, f=f(x). Since our particle was at rest before we
applied this force to it, all the other forces on the particle,
if any, balance each other out and the applied force is the net
or "resultant" force. Now let's consider a tiny
time interval Dt short enough so that
the force is nearly constant during the interval. The average
acceleration of the particle over the interval dt is
a = (v -
v0) / Dt,
where v0 is the initial velocity and
v is the final velocity. The distance x traveled by the
particle during time interval Dt is
the average velocity times the time interval, like this,
x = (v + v0) / 2
* Dt.
The work, w, done on the particle is the force f times the
distance x. The force f though, from Newton's second law is
m*a so the work is
w = m * a * x
or,
w = m * (v - v0)
/ Dt * (v +
v0) / 2 * Dt.
Simplifying the preceding mess,
w = 1/2 * m * v2 - 1/2 *
m * v02.
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Now here is another of those insights that
seem sort of magical. In the expression above for work, w is the
difference between two terms that have identical form. The only
difference in the terms is that v appears in one and
v0 appears in the other. So it looks like the
work done on a particle, which is free to move, during a time
interval Dt, is equal to the change in
the quantity 1/2*m*v2 over that same
interval. If we choose our reference frame so that the initial
velocity is zero, the work done on a free particle shows up as
velocity and the amount of work is equal to the quantity
1/2*m*v2. This quantity evidently has some
close relationship to work.
Suppose now we apply to the particle set in motion in the
preceding paragraph, a force in the opposite direction. During
the application of this force, the particle will experience an
opposite acceleration and slow down eventually to a stop. In this
case the work, w, will be negative since the final velocity is
less than the initial velocity. The magnitude however will be
exactly the same as was the case when the work went to speed up
the particle. Again equal to the change in the quantity
1/2*m*v2. The negative sign on w simply means
that in this case the particle was doing work, rather than having
work done on it. The moving particle apparently has an ability to
do work that the particle at rest did not have.
The ability to do work is defined as "energy" and
the ability of a particle to do work by virtue of its motion is
defined as "kinetic energy". The kinetic energy of a
particle of mass m moving at velocity v is in fact
1/2*m*v2. Remember that
v2 is the scalar product
v·v, so kinetic energy is a scalar even if
we are working in more than one dimension. We will use the symbol
ke to represent kinetic energy. What our calculations have shown
is that the work done on a particle by the resultant or net force
is equal to its change in kinetic energy. This result is known as
the work-energy theorem. If something was at rest in our reference frame and is now in
motion, you may be sure that some work was done on it.
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Our uncovering of the work-energy theorem by working with
Newton's laws is an example of what sometimes happens in
physics. Newton's laws apply only in restricted circumstances
out of the realm of quantum mechanics, very small systems, and
relativistic mechanics, very fast systems. The work-energy
theorem extends to these other realms also, so it is more
fundamental or closer to Nature's truth than the laws we used
to derive it.
Since
work is force, whose units are Newtons, times displacement, whose
units are meters, the units on work are Newton meters. The Newton
meter is given the name Joule, in honor of Mr. Joule I believe
(shown here), so we speak of work in terms of Joules. Kinetic
energy must have the same units as work since the change in
kinetic energy is equal to the work of a resultant force.
I have been making a point that our particle was unconstrained
(free) so that it could respond to the applied force. Let's
look now at a box of rocks of mass m=20kg sitting on the floor.
What is the work involved in lifting it to a height of 1.5 meter.
Assume that the box is lifted with a force of m*g where g is the
acceleration due to gravity, 9.8 m/s2. The person
lifting the box applies a force of 20kg*9.8m/s2 =
196N. The force is applied over a distance of 1.5 meters so the
work of the lifter is 294J. If this were all there were to it, we
would expect the change in kinetic energy to be 294 Joules. But
the box arrives at the 1.5 meter altitude with zero velocity.
What we overlooked in the analysis so far was the interaction
of the box with the Earth. It is true that the lifter did 294J of
work on the box. At the same time, the Earth did -294J of work on
the box, or the box did 294J of work on the Earth if you prefer
that view of things. The work of the Earth on the box was
20kg*(-9.8m/s2)*1.5m. So the net work done in lifting
the box was zero.
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Next let's tie a rope to the 20kg box of rocks and drag it
a distance of 2 meters along the floor. We will begin our
observation of the box as soon as it is moving and drag it
over the 2 meter mark at a constant velocity. The angle that the
rope makes with the floor as we are dragging the box is 30
degrees. The force applied to the rope is 70N. What is the work
done on the box?
The constant velocity condition in the problem tells us that
there is no change in kinetic energy of the box so we know that
the net work done on the box is zero. It appears that the
professor has asked a trivial question. Sometimes we do that you
know. Perhaps the question should have been what was the work
done by the person pulling the rope. The force, f, on
the rope was at a 30-degree angle but the displacement,
r, of the end of the rope was horizontal so the work was
f·r =
|f|*|r|*cos(30) = 70N*2m*.866 = 121.24J .
So the rope puller did work but it did not show up as a change
in kinetic energy of the box and since the elevation of the box
never changed, the Earth did not undo the work of the rope
puller. If that makes you curious about where the work went then
you are thinking the right way about work and energy. Something
must have done negative work on the box to balance the positive
work done through the rope. Remember our old friend, friction?
The force of friction which is always directed opposite to the
displacement would always contribute negative work. The condition
that the velocity was constant tells us that the friction force
exactly balanced the force from the rope, otherwise there would
have been some acceleration. So the combined work of the rope
puller and the friction added up to zero.
What then must have been the coefficient of kinetic friction,
mk, between the box and the
floor? The normal force between
the box and the floor is the weight of the box minus any lift
provided by the rope. The weight is 20kg*9.8m/s2 =
196N. The rope pulls up on the box with a force 70N*sin(30) =
35N, so the net normal force is 196N-35N = 161N. Remember the
magnitude of the frictional force is given by the equation
|F| =
mk * |N| , so
mk = 70*cos(30)N / 161N = 0.3765.
There is a fundamental difference between the lifted and dragged
box situation, which we will explore in the next lesson.
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For now let's look at another example
of work, the work done by a spring. A spring is an object which
exerts a force proportional to its displacement from some neutral
or equilibrium position. The direction of that force is towards
the neutral position. The Work by
Spring display illustrates the effect.
A spring that has been stretched or compressed has had work done on it. The work is stored
in the inter-atomic bonds that hold the spring together and determine its unstressed shape.
When the spring is released, it exerts a force on any object attached to it, as the
inter-atomic bonds in the spring material try to return to their lower energy configuration.
This force times the distance the attached object is moved is the work done by the spring on
the object. As we know, that work shows up as kinetic energy of the object. When the spring
reaches its unstressed length, the object will have some kinetic energy which will carry it
beyond the spring's preferred length, stressing the inter-atomic bonds for another half cycle
of the object's motion.
In the next section we will continue the story of energy,
taking a closer look at the spring and block system.
Are there any questions?
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