The conservation of mechanical energy


Starting with the Newtonian framework for describing motion — objects, interactions, Newton's laws — and asking the question, "what part of Newton's 2nd law tells us how an object changes its speed (irrespective of direction)", we were led to introduce the concept of kinetic energy — ½mv2. The work-energy theorem (Newton's second law for an object, dotted with the object's displacement) led us to introduce the idea of potential energy — energy of place. This led us to see forces of interaction as a way to "store" kinetic energy in a way that it potentially could be restored as a result of the forces objects exert on each other by virtue of their relative location. Thus, the kinetic energy of an object thrown upward is "stored" in the greater displacement of the object from the earth. The kinetic energy of an object approaching a spring is "stored" in the squeeze of the spring. As one positive charge approaches another it slows and "stores" its kinetic energy in its closeness to the other charge.

We define the potential energy as the negative of the work done by the force on the object and move it to the other side of the work-energy equation, treating the effect of a force acting through a distance as a new kind of energy. We were able to do this for three of the forces we have studied: gravity, electricity, and springs. Since we interpret normal and tension forces as a kind of very stiff spring force (see Young's modulus), we can probably define potential energies for these as well (though we usually don't bother for a variety of reasons). What about our other forces? Our resistive forces?

Conservative vs. non-conservative forces

It turns out that we cannot define potential energies associated with resistive forces. We can see this at the macro level by remembering the condition that we needed to turn work into a potential energy: the work done by the force had to be reversible. If it decreased the KE in one direction, it had to restore it when the object moved back. The extracted energy belonged to the place. The marker that allowed us to figure out that this was happening was the fact that the work only depended on the starting and ending positions and not on what happened in between. This way, if you reversed the direction, you would reverse the sign of the work and get the PE back when you reversed. Forces for which this works are called conservative.

This requirement fails for the resistive forces of friction, viscosity, and drag. For two relatively moving objects that exert resistive forces on each other, the forces always point to oppose the relative motion. As a result, the work is negative — the force and displacement are in opposite directions (and cos 180° = -1) so it devours KE. If you try to reverse the motion, the forces reverse too and the work is still negative. It continues to devour KE instead of restoring it. Forces like this — whose work can't be written as a change in a potential energy — are called non-conservative.

The simplest example is a block sliding on a table. If I give it a push the block will slide, but the friction from the table will slow it down and bring it to a stop. If I push it back the other way, the friction will still slow it down — not speed it up, restoring the energy it stole. So the work done by these kind of forces cannot be written as a change in a potential energy.

A work-energy theorem with PE

Oh well. The PE made things nice and gave us a good way to think about motion, but we can still write a work-energy theorem by separating the conservative and non-conservative forces. We write the work done by conservative forces as a PE and leave the work done by non-conservative forces as work.

$$\Delta \bigg({1 \over 2} \;  m v^2  + U_{gravity} + U_{springs} + U_{electric} \bigg) =\overrightarrow{F}^{net}_{non-conservative} \cdot  \overrightarrow{\Delta r}$$

This is, of course, only the energy equation for a single object and the PEs are assigned to it assuming that the other objects causing our object to feel conservative forces don't move significantly. If they do, we would have to add together the energy equations for the two objects and only assign a single PE since it really belongs to the interaction, not to an object.

A conservation law

This equation does provide us with a conservation law and the conditions when it can be used. If we have one object interacting with other fixed objects we get the following:

The principle of conservation of mechanical energy (one object): When an object feels conservative forces from other objects whose position can be considered fixed, and when the resistive forces felt by that object can be ignored, the work-energy theorem takes the form of a conservation law:

$$\Delta \bigg({1 \over 2} \;  m v^2  + U\bigg) = 0$$


$${1 \over 2} \;  m v_i^2  + U_i = {1 \over 2} \;  m v_f^2  + U_f$$

where "i" labels the variable in the initial state, "f" labels the variables in the final state, and U represents the sum of all potential energies — gravitational, electric, and spring.

If we have a system of interacting objects that interact with conservative forces only,* we can add together the individual work-energy theorems. (We have to be careful to only count the interactions between a pair of objects once.) The result is

The principle of conservation of mechanical energy (many objects): When a set of objects, i = 1,...N interact through conservative forces, and when the resistive forces in the system can be ignored, the work-energy theorem takes the form of a conservation law:

$$\Delta \bigg[\sum_{i=1}^N \bigg({1 \over 2} \;  m_i v_i^2\bigg)  + \sum_{j=1}^N \sum_{i=1}^j U_{ij}\bigg]= 0$$

where $U_{ij}$ represents the sum of all potential energies — gravitational, electric, and spring — between the objects i and j.

When is it useful?

This way of describing motion as an exchange between kinetic and potential energies is particularly useful when the conditions of the principle are met and when we don't care about the time. If you're sliding down a (nearly) frictionless slide and want to know how fast you are going when you get there — but don't care how long it takes you — the conservation of mechanical energy will tell you. If you throw a ball straight up at a certain speed and want to know how high it goes — but don't care when it gets there — the conservation of mechanical energy will tell you.

Basically, the conservation of mechanical energy is useful whenever you want to find how position and velocity are correlated and you can ignore resistive forces.

What's really going on?

What is this really about? We know that total energy is conserved. How is this theorem about "mechanical energy" different? What's going on with our resistive forces?

Although we know energy is conserved once we have defined all possible forms, not all forms of energy are equally useful. In all our discussion of Newton's framework for motion we have been talking about moving of objects. Since we know objects are made up of molecules, what this tells us is that all the object's molecules are moving coherently. This is what we mean by mechanical energy. When friction or drag drains the kinetic energy of a macroscopic object, what it's doing is converting the coherent mechanical energy of the object into the incoherent random motions of the molecules of the object. When an object's KE is drained, all of its molecules may be going faster (its temperature goes up a bit), but they are going every which way. It's not easy to make those motions coherent again.  In the follow-ons we'll study what happens to the lost energy (thermal energy) and when it is possible to recover it as coherent macroscopic motion (free energy).

* This process is trickier than it sounds and harder than in the case of generalizing momentum conservation to many objects. This is because the PE involves the relative positions of the objects so a tricky coordinate transformation is needed before the PEs can be correctly constructed.

Joe Redish 11/5/11


Article 443
Last Modified: February 21, 2019