Can internal forces do work?

Question

My Mechanics textbook claims that the sum of the work by internal forces is not generally zero.

translated to English the paragraph reads:

Notice about the work by internal forces: the work by the internal forces is in general not equal to zero although the sum of the internal forces is always equal to zero. Since the work also depends on the relative displacement of the two pointmasses in the material system, the work provided by these 2 internal forces will generally not be equal to zero.

Other sources claim that the work done by internal forces is always zero. Why does a car engine not do work if the wheels don't slip? (answer by Mark Eichenlaub)

Then I didn't do any work on the system at all since, by definition, only external forces can do work.

http://in.answers.yahoo.com/question/index?qid=20110131130859AAMLJLd

the net work done by internal forces is zero since they do not cause motion of the object.

To me it would seem that these sources contradict eachother. Do they? If so, which one is correct? If not, please explain why they don't contradict eachother.

Answer 1

Well, I think the issue must be that all those references that claim that internal forces don't do work must refer to RIGID BODIES or PARTICLES. If that's the case, they're right. But in general, for deformable bodies, that's not true and your textbook is right.

• In a rigid body, all the external work done to the body is transformed into kinetic energy. The work of the internal forces is zero, since any couple of internal forces will act in opposite directions (the former fact is known as the work-energy theorem)

• In a general body (think of continuum mechanics), part of the external work will be done to compensate for some internal work, and not all of it will become kinetic energy. For simplicity, let's assume that the internal forces come from a potential and let's see an example:

Let our system be a spring that is compressed by an external force F, until it reaches a steady position. There is an external work on the system because the force creates a displacement, but in the end the spring is steady, so there is no increase in the kinetic energy between the initial and final states. Where did all this energy go? It compensated the work of the internal forces of the spring. Because these internal forces derive from a potential, the work of the internal forces amounts exactly the change in the potential.

In general we have, with usual sign conventions (but that changes a lot, it is better to think carefully on the problem than to just substitute) $$ - W_e = dK - W_i = dK + dU $$ Where $W_e$ is the external work, $W_i$ the internal, $dK$ the change in kinetic energy and $dU$ the change in the internal energy, that in the example is just the change in the potential.

Answer 2

Work is a subtle concept that can be approached in various ways. There are ambiguities that can be reduced somewhat through the careful use of language. Ultimately, however, these ambiguities can only be eliminated by specifying a theory of physics and stating the definition within that theory.

Suppose our theory is Newton's laws applied to frictionless gear trains. Then the forces are normal forces exerted by a gear tooth A on another gear tooth B. By Newton's third law, the forces are equal in magnitude and opposite in direction. Since these are normal forces, the displacements of A and B are equal. Work can be defined either as $dW=F\cdot dx$ or as the transfer of energy by a mechanical force; since these are normal forces, the two definitions are equivalent. By Newton's third law, A's work on B cancels B's work on A.

Suppose instead that our theory is Maxwell's theory applied to oppositely charged particles A and B moving in a vacuum. We release A and B at some distance from one another. Their subsequent motion is that they oscillate about their common center of mass, radiating electromagnetic waves (and periodically passing through each other). There is clearly no way we can define work so as to make A's work on B always cancel B's work on A. Energy is going out into the electromagnetic waves, and there is no way to count this energy into the definition of work, since it's nonmechanical.

Answer 3

Changing our notions of the universe has always been a challenge..."The sum of all internal forces is not zero. If the sum of the internal forces is zero, the universe not would be dynamic"

Is known of the mechanics that when a body exerts a force of field towards another material body, the same body receives equal force and contrary to that exerted, and this is how it is understood the law of universal gravitation formulated by Newton, this that well is true for bodies that possess inertial values similes in magnitude stops being necessary when used bodies whose inertial magnituds may vary in thousandfold the value of the smaller body, for saying a number. This is the case of the system formed by the Earth and an object, as is a tennis ball, this will receive variation in its quantity of motion to the moment to be put down in free fall, but it is not possible to say the same of the Earth, in this case the variation in its motion is null in practice, since this has not moved thousandth none of distance and less to experienced some type of acceleration. Now analyzing the existence of such repose and displacement of the object, is true to say that the body higher energetic content exerts field influence to the interior of the body that we consider as the system to studying.

Then, a body is considered in state of rest when the sum of all the forces acting in it is zero, and is also applicable to situations in the empty space when the field of predominant acceleration is slightly influential for a given time interval, what it wants to say that in the charted of a continuous curve, rather it approximates to a straight line. However a body free of rotation for a given time interval, is considered in a state of absolute rest when in it no external forces are acting or influences of field, so is this who exercises the predominant field on other material bodies that compose a system, thereby generating the point of reference for obtaining an absolute space, is to say an euclidean space. Furthermore, expanding our field of vision, in any system the sum of the internal forces never reaches to be zero, though the body is in equilibrium, what must not get confused to idealize a isolated system, that in the practice, as this way the experience affirms, is not possible to find a system with such characteristics, due to that not exist some material that may isolate the gravitation. Is for this reason that the interaction between systems is the key for understanding of the Principle of Internal Force, which is capable of solving a series of problems own of the mechanics and the electrodynamics.

Answer 4

No. "W=0 for internal forces" is valid for all bodies irrespective of their rigidity. Because a non-rigid body can be deformed only due to external forces, not due to internal forces, because internal forces act along the same line. Since they are equal and act in opposite directions, this causes the net force to be zero.