PHYSICS 1001 REGULAR 2001
Lecture 4
WORK - KINETIC ENERGY
Copyright J.B.T.McCaughan March 2001
PREAMBLE:
The most dangerous law in physics:
F = ma
A formal relationship connects CAUSE (Impressed Force F) to EFFECT
(A body of mass m accelerated by a).
But there are four categories of cause: FORMAL, MATERIAL
EFFECIENT and FINAL. The formal relationship (mathematical
equation) is to nature as a plan is to a building:EXTRINSIC
FORMAL CAUSE. In the case of the building, its shape is the
INTRINSIC FORMAL CAUSE, which should closely reflect the
shape in the plan. The builder causes, through the efficient cause,
the building to be built from brick, tile, wood etc - the material
cause. Why the building was built, the final cause, is first in
intention and last in excecution. E.g. to have a house for my
family, to rent out for profit, to house a business etc. In scientific
circles final cause is shunned. Aristotle turned nature into an
organism and gave final cause the dominant role. Things moved
because they sought their place. The outcome of this strategy is
to provide answers to WHY things behaved the way they did,
not HOW they behaved. One could then be content to
contemplate nature not figure out how it worked and how it
could be harnessed as we now do. Nevertheless final cause can
still be found operating in physics.
The reason that the second law is dangerous is that the (extrinsic)
formal cause of the equation refers explicitly to the efficient cause. It
appears that the efficient cause is subservient to the formal cause of the
equation. We apparently have a license for believing that the equations
can somehow reach out and move the nature they are describing. A
license for believing that the equations control nature. This very
common view is mistaken. What the formal cause incorporates is the
measure of the force, not force itself. The measure has mental
existence, not real existence, since measures were set up by the mind. It
is a mental existence based on the real in that the thing chosen to be the
measure is material and therefore real; that it is a measure is mental.
This change of mode of existence renders force suitable for
manipulation by formulae, which have the same mental existence. But
measures (mental existence) are not the things themselves (real
existence) whose measure they are. So the equations do not control
nature, they describe it.
So far the only mathematical law we have for Dynamics is Newton's
second law. It is the foundation law for the rest of mechanics. But it
only tells us how much bodies accelerate in response to a force. We
need to know what happens to these bodies after they have been
accelerated for some time and for some distance; what happens to them
in time and space.
AS FAR AS MECHANICS IS CONCERNED THERE IS NOTHING
MORE IN NATURE THAN MATTER IN MOTION SUBJECT TO
INFLUENCES CALLED FORCES, THE REST IS BOOKEEPING.
The one and the same motion in nature can be bookept from the point
of view of KINETIC ENERGY or MOMENTUM. These separate
concepts, derived from considering the action of one and the same
force over distance and time, refer to one and the same motion. Kinetic
energy is the measured effect of WORK (the measure of force over
distance). In nature there is only a change in the state of motion under
the influence of a force. It can’t help but act in space and time
simultaneously. Momentum is the measured effect of IMPULSE (the
measure of force over time).
WORK: Work is done BY a force when it moves its point of application
in the direction in which it acts. (If the point of application moves in
the opposite dircetion to which it acts, then the work done by it is
negative).
This is the condition that has to be observed in nature for work to be
ascribed to the agent force. This condition has real existence. It is not
mentioned in the text books these days, the formal definition (below)
only is given; a tacit approval for the mathematics controlling nature?
x2
 F  dx
W=
x1
W   F cos  dx
F
O
dx
The expression to be integrated is the scalar product of two vectors.
This is one form of the product of two vectors, the other you will meet
is the vector product. As the name of the product implies, Work is a
scalar. Hidden in the dot product of the equation in the first line is
cosine of the angle between the directions of the two vectors. This is
rewritten in the equation of the second line and illustrated in the
diagram.
WORK - KINETIC ENERGY THEOREM:
If there is only one force acting on an object, then the work done BY the
force NECESSARILY results in an increase in kinetic energy of the
object. This is DEDUCED from the expression for work and the effect
of one force in NII. NII is the foundation of work - kinetic energy.
dv
dv dx
F

dx

ma
dx
=
m
dx

m


 dt
 dx dt dx
  mvdv 
12 mv2 v
v2
1
One must be careful to distinguish between work done ON and work
done BY. It is quite possible for the work done BY a force not to equal
the increase in kinetic energy of the object on which it acts.
DEMONSTRATION: Duster pushed at constant speed across the desktop.
It is obvious that the conditions for the agent (lecturer) to do work are
fulfilled. The finger applying the force to the duster moves its point of
application in the direction in which it acts, the agent has obviously
done positive work. Yet there has been no increase in the kinetic
energy of the duster. Clearly there is at least one more force acting on
the duster, viz. friction from the table-duster surface. The work done
by friction is the negative of the work done by the agent. The point of
application of friction moves in the opposite direction to its
application. The NET work done ON the duster is zero.
Now we already knew that the net force on the duster was zero as
there had not been any change of state of the duster. So if the net force
is zero there can’t be any work done on it. We didn’t need to go
through the calculation of the work done by either force to reach that
conclusion, but it shows the consistency of the definition.
COMMENTS:
(a) HRW Chapter 7 (6th edition) has opted for an incoherent approach
to work-kinetic energy. It defines kinetic energy first out of the blue
with no reason for why it has the form it has. Worse still it claims
‘Work is (kinetic) energy......’ (p118).(This is a total confusion between
cause and effect. Work is NOT identically equal to kinetic energy or
any other type of energy. The measure of work is not the measure of
kinetic energy. The measure of work may not even equal the measure
of kinetic energy as we have just seen. Work and kinetic energy have
separate existences based on the real: On the side of the agent force for
work and on the side of the object undergoing motion for kinetic
energy.
(b) The example of the duster and agent force above raises the question
as to happened to the work done by the agent? Did it go for nothing,
was there no change? The agent brought into play friction, which in
turn generated heat detected as a rise in temperature of the duster and
to a lesser extent the surface. This is an entry point for the science of
heat. We pay no further attention to it in mechanics.
DEMONSTRATION: Duster raised vertically at constant speed.
This demonstration is similar to the last one in that work is done BY
the agent force, yet no increase in kinetic energy of the object results.
There was no net work done ON the object since there was no net force
acting on it. Gravity was doing negative work.
The question can be raised again as to what happened to the work
done by the agent. Did it go for nothing, was there no change? Yes
there was a change: the earth and duster have been separated. If the
agent is now removed, gravity has been primed so it can now do
positive work: The duster falls gaining the increase in kinetic energy it
would have got from the agent if the agent had acted alone (assuming
no loss to air friction on the way down). The work done by the agent
appears to have been stored in the gravitational interaction between
the earth and the duster (what we call the gravitational field). Since
this stored work later appears as energy it is called POTENTIAL
ENERGY. We return to this next lecture, the remainder of this one is
devoted to the technical exercise of computing work in the
gravitational context.
y2
F
W
y1
By NII the equation of motion is:
F-W=0
We now consider in turn the work done by the agent force and the
work done by gravity.
WF 
y2
y2
y1
y1
 F  dy   F cos 0 0 dy  Fy 2  y 1
y2
WW   W  dy   W cos 180 0dy  W y 2  y 1
y1
Since F=W the NET work is zero on the duster.
Now the duster has been raised to y2 the agent force is removed and
the duster falls. Formally this is treated as follows:
y1
y1
y2
y2
W W   W  dy   W cos180 0 dy  W y1  y 2 
W W  W y 2  y1 
This is now a positive quantity. NOTE that the angle remains 180
degrees. Formally, once the positive direction is chosen it remains
fixed. The direction of W remains opposite to the positive direction
whether the duster is raised or lowered. However if one considers
what happens in reality, not formally, one would say that the force
moves its point of application in the direction in which it acts
(downwards). This must mean that positive work is done BY the force;
which confirms the formal treatment. The danger is to mix the formal
and real approaches and convert the angle to zero degrees.