On Work and Energy

Question

I saw in my class that work done on an object is the energy transferred by a force. Energy is the ability to do work.

question 1) Why is this definition so roundabout?

question 2) How exactly does a force transfer energy?

P.S. (I am comfy with single variable calculus)

Answer 1

1. "work done on an object is the energy transferred by a force. Energy is the ability to do work."

This is indeed a circular argument. It won't be if we replace the first sentence above with a definition of work...

The work done by a force of magnitude $F$ moving through a distance $d$ is defined by $W=Fd \cos \theta,$ in which $\theta$ is the angle between the force and the displacement.

This definition, together with the definition of the energy of a system as the work it can do, will enable you to derive potential energies of bodies in uniform and inverse square law fields, to derive the kinetic energy formula and, together with the law of conservation of energy, to solve all sorts of problems in dynamics. Arguably, the definitions need modifying or, at the very least, re-interpreting, for use in thermodynamics and quantum mechanics.

2. How exactly does a force transfer energy?

Maybe looking at individual cases will help. (a) Imagine you apply forces to a spring and stretch it, so work is done. The spring can do work when released (for example accelerating a toy missile), so you clearly transferred energy to it. (b) You push, and keep pushing, a sledge on level ice. It gets faster and faster, acquiring kinetic energy. We know that when it is moving it has energy, because it does work as it slows down, for example we can tie a rope to it and make the rope turn a generator and provide electrical energy for a short while.

Answer 2

question 1) Why is this definition so roundabout?

In my opinion, I don't think the definition is roundabout. The fact that the definition of energy seems roundabout has more to do with the fact that energy is continuously being transferred between objects or systems that are not isolated from one another (i.e., objects not having boundaries that prevent both work and heat transfer as well as mass transfer).

Energy can be considered to be a property that an object possesses. The object’s internal energy is the sum of its kinetic and potential energies at the molecular level, as reflected in properties of the object such as its temperature (for kinetic) and the intermolecular forces (for potential). This is generally the realm of thermodynamics.

The objects external energy is the sum of its kinetic and potential energies at macroscopic level, i.e. with respect to the mass as a whole, and is determined with respect to an external frame of reference. Examples are gravitational potential energy and the kinetic energy of the object as a whole due to its position or velocity, respectively, with respect to an external frame of reference. The external energy is generally in the realm of mechanics.

Work is one of two forms of energy transfer between objects. The other is heat. The former is generally in the realm of both mechanics and thermodynamics. The latter is generally in the realm of thermodynamics.

If an object possesses energy it has the capacity to do work. What this means is that it has the capacity of transferring some energy from itself to some other object by applying a force through a distance, resulting in a change in the energy (internal and/or external) of itself (decrease) and the other object (increase), based on conservation of energy.

question 2) How exactly does a force transfer energy?

I believe the examples given by Phillip Wood answer this. I will add another simple example in the context of my answer to question 1.

I am sitting still at my desk with my feet on the floor. Portions of my body possess external potential energy with respect to the floor depending on the product of their mass and their height with respect to the floor. Since I am sitting still, I have no external kinetic energy with respect to the frame of reference of the room I am in.

I have internal kinetic and potential energies (at the molecular level) as reflected by my body temperature and intermolecular forces. These internal energies are due to conversion of chemical energy by mean of metabolic processes. If I am sitting still there is no work transfer between me and other still objects in the room. Since my body temperature is greater than the room temperature, there is heat transfer from my body to the room.

I now decide to take an object from my desk and put it on a shelf above the desk. To do this I must apply a force to the object against the force of gravity to lift the object. That force times the increase in height of the object is energy transfer by work from me to the object. That work increases its gravitational potential energy at the expense of my internal energy. By virtue of my increasing the gravitational potential energy of the object that was on my desk, I have increased that objects capacity to do work, given the opportunity and circumstances to do so.

Hope this helps.

Answer 3

1. Work done on an object is the energy transferred by a force.

2. Energy is the ability to do work.

I agree that these two sentences look like a roundabout. However. coming to your question n.1, no one of them is an acceptable definition of work or of energy.

One has to be careful about sloppy definitions even if present in textbooks or in web-sites about physics. Sometimes (in some cases, very often) physicists and teachers use informal or pictorial sentences which should convey some important idea about a concept. It is not a bad practice. But it should important to make clear the border between a real definition (which shouldn't be circular) and sentences which are there just to emphasize a concept, but cannot be a substitute of a clean definition. That's the case of the above quoted sentences 1 and 2.

Before discussing them, let's see what is the correct sequence of definitions. In the following I'll assume that the context is the classical mechanics. However a similar approach applies also to thermodynamic relations between work and energy.

In Newtonian mechanics, energy is not a primitive concept. It is introduced, and its importance is evident, through the work-energy theorem. By examining the proof of the theorem, it is evident that the concept of work is a prerequisite for this theorem, which introduces the fundamental concept of total energy (kinetic + potential).

Thus, the correct conceptual sequence is $$ Force \rightarrow Work \rightarrow Energy $$ Said more explicitly, work does not require the concept of energy for its definition and indeed the Classical Mechanics definition of the work of a force ${\bf F}$ to displace a point-like particle on a path $\gamma$ is: $$ W_{\gamma} = \int_{\gamma} {\bf F} \cdot d{\bf r}, $$ which is a line integral requiring only the force as dynamical ingredient. No energy appears in this definition.

Actually, the (total) mechanical energy is defined in two steps.

1. definition of kinetic energy: a quantity depending only on the mass and square modulus of the velocity ($\frac{1}{2}mv^2$);
2. the definition of potential energy difference when the forces are such that their work depends only on the initial and final position and not on the path $\gamma$ ($\Delta U = W_{\gamma}$);

at this point, the work-energy theorem allows to introduce a new quantity (total energy = kinetic energy + potential energy) which is a constant of motion.

Why 1. is wrong as definition of energy? Because, as discussed above, we do not need energy to define work. The only way of keeping sentence 1. is to look at it not as a definition but as a statement expressing in words a key point about potential energy. However, this is a result, not a definition.

Statement 2. is wrong, in general. By inspecting the definition of energy it is clear that potential energy is defined in term of the work done by the force on the mechanical system, not by the system. And in general there is no direct relation between these two quantities one could derive from the equations of motion. The sentence can still be used to convey one of the most important features of energy: its ability of being modified from one form to another. But it should be stated with some words of caution:

1. first of all, for every mechanical system, it is not the energy which matters but its difference with respect to the minimum. If a system is at its lowest energy configuration, no work can be obtained from it;
2. moreover, there are situations where the identification of the system which has energy and does work may be non trivial. For example, let's consider a fixed charged plane and a particle of the same sign arriving on the plane from the orthogonal direction. It has a well definite kinetic and potential energy. Changes of potential energy are obviously connected to electric work. However, the moving particle does no work at all. The work is done by the electric field which modifies the potential energy of the particle. Of course, one could say that in this case the potential energy is not a property of the particle alone but of the whole system (charged plane + particle). But this is just a different way to stress the problems connected with statement 2.

Answer 4

question 1)Why is this definition so roundabout?

There are many different definitions of work and energy. In mechanics energy is “the capacity to do work” and work is “$W=\int F\cdot dx$”, which is not circular. In thermodynamics energy is “anything that can be converted into $KE=\frac{1}{2}mv^2$” and work is “a transfer of energy through any means other than heat”*, which is also not circular.

The circularity comes from using the mechanics definition of energy with the thermodynamics definition of work. So you should avoid doing that.

*This is not mainstream terminology but I personally wish that the thermodynamics definition of work was simply “a transfer of energy” and then heat would be “thermal work”. Alas, I am centuries too late to cast my vote.

question 2)How exactly does a force transfer energy?

A force doesn’t necessarily transfer energy, it transfers momentum. In order for a force to transfer energy/do work the object on which the force is acting must be moving. The power, which is the rate of energy transfer, is $P=F\cdot v$ because $W=\int F\cdot dx=\int F\cdot v \;dt=\int P\; dt$ So, to answer your question, how a force transfers energy is by acting on a moving object.