Lecture 8

Work and Energy

Energy is probably the most important concept in Physics as well as in other Physical Sciences. In our current understanding, it wouldn't be far from the truth to state that Everything is Energy. Another equally important concept, in all Physical Science is the principle of Conservation of Energy. Even though a postulate that cannot be proved absolutely (nothing can be proved absolutely in Physics), assuming its validity is one of the most useful tools we can exploit. Before now you have certainly encountered the term "energy" many times, and you have heard of many different types of energy (nuclear, electrical, chemical, potential, thermal, etc.). We will eventually study them all but, for the time being, we will limit ourselves to the study of Mechanical Energy, i.e. energy related to the motion ( or possibly the location) of the bodies under exam.

In spite of its importance, it is not straightforward to define energy. Moreover, the concept of energy has evolved with time, following the progress of Physics knowledge.

Another problem : energy is even less clearly defined in the common language. If you look into a dictionary, you might see that Energy, Power and Force are often used interchangeably. This is not so in Physics, and it is extremely important to understand the proper definitions and the clear distinctions between these three quantities.

Let us then start with the concept of Work. In physics, work is given a more restrictive, but more precise, meaning than what we encounter in everyday's language. The idea of (Physics) work is associated a Force acting on a moving object. We will say that a force does a certain amount of work on an object, when it acts on it while the body moves over a certain distance. In the simplest case, when force and displacement are along the same direction, we will say that when a force F moves an object over a distance d, it does an amount of work W given by

$W = F \times d$

From this definition we can see that Work will be measured in Newton x meter and this unit is called the joule (J). It is a very important unit since, as we will soon see, it is also the unit for energy.

In the more general case, where the force and the displacement do not have the same orientation, we generalize the definition above by

$W = F \times d\cos\theta$

where $\theta$ is the angle between the displacement and the force. From this definition we can infer some extremely important consequences :

1.

when force and displacement are parallel to each other, the whole magnitude of the force contributes to the work. If they are not parallel, only the component of the force in the direction of the displacement does work.

2.

when force and displacement are perpendicular to each other, NO WORK IS DONE by the force

(standard quiz question : a satellite circles the earth in a perfectly circular orbit; what is the work done by gravity?)

3.

if the angle between force and displacement is between 90⁰ and 270⁰, the work done by the force is negative (Work is a scalar, i.e. a number, and it can equally well be either positive or negative).

Food for thought : compare the situation of an object moving with constant velocity on a frictionless surface, with that of an object being dragged (at constant velocity) along a surface with friction. Who is doing what work (if any) in either situation?

Answer Conceptual Questions 6.1,2,3,4 . Let me know if you have a problem.

Problem 6.12

Another interesting example : weight lifter bench-pressing a barbell. The book discusses the work (and its sign) done by the muscular force. What about the force of gravity?

Energy

Now that we understand Work, we can define Energy : Energy is the ability to do Work. Quantitatively, we can actually state that the energy of an object is given by the amount of work it can do, which tells us that Energy will also be measured in joules. All this is a bit vague, but its meaning will become more and more clear as we proceed. To start, we will study the connection between Work and Mechanical Energy. Let us suppose that we have a net, non-zero, force acting on a body: from what we have learnt, this will cause the body to accelerate (notice that this is different from the case we looked at before, where we were assuming constant velocity, therefore no net force; in those cases, equal and opposite forces were also doing equal and opposite work,therefore no net work was being done, even though each force by itself was doing some work). So, if we have a net force F causing a body to move a distance d, the work done by the force is

$W = F \times d$

But, according to Newton's Law, F= ma, therefore

$W = m\times a\times d$

By effect of the acceleration the body changes its velocity. The expression we had found (one of the ones I had asked you not to memorize) relating acceleration, change of velocity and distance was

v² = v₀² + 2ad, i.e. ad = (v² - v₀²)/2

Substitution gives

W = m(v² - v₀²)/2 = 1/2 mv² - 1/2 mv²₀

This result represents the Work-Energy Teorem. If we call 1/2mv² the Kinetic Energy, we can say

The effect of a net force doing work on an object is to change its kinetic energy from the initial value to the final value

One way to look at this is to say that work is not wasted, but it is "compensated" by a growth in energy (note that this growth does not contradict the principle of energy conservation, nor is energy being "created" by work. The correct statement, as we will see better in the future, is that work has the effect of transporting energy from one form and/or object to another).

It should be immediately clear that if the net force is doing negative work on an object, its kinetic energy will accordingly decrease.

In some of the previous examples we had a body moving with constant velocity; in those situation there was no net force, therefore no net work, but also no change in kinetic energy ($v_{initial}=v_{final}, \Delta E_{kin} = 0$), therefore everything is consistent.

Problem 6.22, old and new procedure