© Timothy E. Chupp, 1995

We now have a feel for energy and for conversion of one form of energy to another
(for example from chemical energy to heat). Work is the energy transferred from
an agent applying a force to an object or into heat, and the ** net** work done
on an object is the
total change of kinetic energy. Both positive work and negative work have
meaning:

Positive work follows when the force has a component parallel to the displacement. Positive
work adds energy to a system.

Negative work follows when the force has a component opposite or against the displacement.
Negative work removes or dissipates energy from the system.

Two examples: In pulling a box of books along a rough floor at constant velocity,
I do ** positive** work on the box, that is I put energy into the system. The
force of friction opposes the displacement (always), and does ** negative**
work. The ** net** work is the total of my positive work and fiction's
negative work, * i.e.* ZERO. Thus the Kinetic energy of the box does not
change: its velocity remains constant.

Another experience: When pushing a car to speed it up for a jump start, we
push on the car in the direction it is moving, that is we do
** positive** work. It is also possible to slow a car down by pushing
on it. In this case we push opposite the direction of motion. We do
** negative** work, which removes kinetic energy from the system.

Consider the simple task of lifting a book from the floor to
a convenient height, say to a 1 meter high counter top. The process
involves first accelerating the book * upward* from rest 0.1 m, then moving
it with a constant upward velocity for 0.8 m, and finally accelerating the
book * downward* (that is decelerating it) to bring it to rest 1 m above the floor.
It is convenient to show this task on a graph of the applied force versus displacement. This
is now a very meaningful graph on which the area "under" the curve
is a measure of the work.

Figure 1: A graph of applied force vs displacement for the task of lifting a 1 kg book a total of 1 m above the floor.

Let's discuss several aspects of this graph: Over the first 0.1 m, the book
is accelerating upward, which requires an imbalance of the upward applied
force (12 N) and the downward force of gravity (-10 N). The work done
by the applied force is 12 N x 0.1 m = 1.2 J. The ** net**
work done is (12 N - 10 N) x 0.1 m = 0.2 J. This ** net** work
is the change in KE of the book. ( so ). The
work done over the 0.8 m from 0.1 m to 0.9 m is 10 N x 0.8 m = 8 J,
but the net force is ZERO so the net work is ZERO. Over the final 0.1 m, the
applied force of 8 N does 0.8 J of work, but the ** net** work is
(8 N - 10 N) x 0.1 m = -0.2 J, that is the change in KE is
* negative* 0.2 J bringing the KE and velocity to ZERO. The
total area under the applied force vs velocity curve is 0.2 + 8 + 0.8 = 10 J, and
the net work, that is the net change in KE, is ZERO.
This would be the case for any combination of acceleration and deceleration.
Try it.

Now that the book is at the height of 1 m above the floor, having had some
work done to get it there, the energy used can be converted to kinetic energy
by dropping the book. When the book is dropped, its velocity and **KE**
increase as it falls until it hits the floor with a bang or a splat. In the
process of hitting the floor the Kinetic Energy is converted into other kinds
of energy such as heat, sound and perhaps some work that permanently deforms
the shape of the book. Before the book is dropped, the work done in lifting
it is stored. This stored energy is available for conversion into **KE**, heat energy,
sound energy, etc. We often call this stored energy ** Potential Energy** or
position dependent energy: **PE**.
It is also properly called gravitational potential energy (to distinguish it from
electrical potential energy) and, as we'll define it soon (Lecture 16), in terms of
the change of gravitational ** binding energy**.

The stored **PE** is given by the total work done in lifting the book, that is by the
total area under the curve of applied force vs position. To be exact, we say

Only the change of energy, and the change of **PE**, make sense. However, we often
define a certain position as our reference point, * i.e.* **h=0** with
**PE=0** at **h=0**. Adopting this convention,

For example, the floor is at **h=0**.

The work done in lifting a book can be stored as **PE** and then recovered. This is true
no matter how the book was lifted, that is if it was accelerated and decelerated
rapidly or slowly. In principle, we could lift and horizontally translate the book
so that it is stored at a height **h**, but with a horizontal displacement. Since
the book must first gain and then lose some horizontal velocity, first positive and
then negative work is done
on the book by who or what moves it. In many practical cases, the positive work
is not actually "recovered," that is it is not stored in the same way that gravitational
**PE** is stored. This is why it is often emphasized that "the book is moved horizontally
** very slowly**," so that the extra work done is so small as to be negligible.

In cases for which the work done by an applied force can be completely stored as **PE**, we say
that work is done ** against a conservative force**. In the case of the book, the conservative
force is gravity. Electrical forces are conservative, but frictional forces are not.

We will consider a large number of examples, including that of lifting the book, that
demonstrate the Conservation Law of Mass/Energy. This law is best stated that
any system has a total energy **E** that remains constant. The total energy is the
sum of **KE**, stored energy (**PE**), heat, sound, etc. In fact heat and sound and
many other forms of energy are manifestations of motion, that is of the molecules
of an object and of air. The contribution to **E** will be discussed more
in Lecture 16. Conservation of energy can be expressed as follows:

A long massive pendulum is a good example of conservation of energy because the amount
of heat energy generated each cycle can be quite small compared to the maximum
**KE** and **PE**. Focault used a pendulum, with a special pivot, that always swings in the same plane
so that one can observe the earth turning underneath the pendulum swing during many
hours, even days. As you observe the pendulum swing, you can watch energy converted
from **PE** to **KE** to **PE** etc. as shown. If you watch long enough or there is appreciable
friction, you will discern that energyy is also converted into heat.

Neglecting this production of heat, that is neglecting friction, we can write the total energy and conservation of energy as

Where **h** is measured above the lowest point. Note that the maximum **KE** occurs at
**h=0** and the maximum **PE** occurs where **v=0**, that is

Figure 2: The pendulum Bob at three positions. The **KE** and **PE** are indicated at each position.

Let's explore this example a bit further. To do so, we consider the free body diagram for the pendulum Bob including the tension in the pendulum wire and the gravitational force on Bob. This leads to:

At the lowest point of the swing, when the velocity has its maximum magnitude, the acceleration is centripetal, and directed straight up with magnitude

Substituting and noticing that ,

We can solve for the tension

Thus **T** the tension in the pendulum wire depends on the velocity, and is maximum when the
**KE** is greatest, that is at the lowest point of the pendulums swing.

An interesting demonstration of this, and thus a way to confirm our application of conservation of energy to the pendulum Bob, is the set-up with a 1 kg Bob and a second, 2 kg mass (Pat) connected to the pendulum wire by way of a pulley as shown.

Pat will inform us when the tension in the string is greater than (the product of Pat's mass and the local gravitational field). This happens when

With the help of a little trigonometry, you can see that

And when .

When we tried this in lecture, we found that was not quite enough, because it is not true that "no energy is lost due to friction," even for a half-swing of the pendulum. When we made the angle slightly greater than , Pat was lifted a bit.

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