What is the difference between applying force and spending energy?

Question

I have a follow-up to this question on how objects can be held up, against gravity, without work being performed. This comment is what I get tripped up on:

Unlike your arm, the table does not need to spend energy to hold up the book.... All the table needs to do is apply Force, but applying force does not necessarily mean spending energy.

Should I then expect holding up a bowling ball to make my arm no more tired than holding up a book?

Or asked a different way: If I place a very heavy stone on a wooden table, and return after a month, to find that the table has deformed, does that prove that the table has "spent energy" holding up the stone, and that this energy loss is somehow reflected in a compromise of material integrity?

Answer 1

If a force is performing work, then it is causing some kind of change in the thing receiving the force: position, velocity, temperature, shape, etc. If a force is causing no changes, then it is doing no work (see the answer here).

In the case of the table, placing any weight on top will cause it to deform a like a spring. After this initial deformation (which does require work to be done on the table, since it changed), there are two possible end results. First, the weight is light enough that the table can hold it up indefinitely without further changes in shape. Second, the weight is heavy enough that the table continues to deform and sag to the ground before breaking (which may be a fast or slow process). The first situation is called elastic deformation since the table will return to its original shape one the weight is removed. The second situation is called inelastic deformation. Removing the weight will no return the table to its original shape (especially if it's broken).

Whether the deformation is elastic or inelastic comes down to whether the weight is large enough to overcome the forces holding the table together, namely the bonds between the atoms that make up the table. Elastic deformation only stresses and stretches these bonds. Inelastic deformations break these bonds, resulting in permanent changes to the structure of the table. If the table continuously sags due to the weight placed on it, that is evidence that the structure of the table is changing, and that the force is doing work on the table. If the table is holding steady, then no work is being done because nothing is changing.

In the case of your arm, as described in this answer, the structures that cause your arm to hold up a weight are temporary by nature. Thus, to hold up a weight, the structures that cause your muscles to contract have to be constantly reformed. It's a little difficult to tell that work is happening because your arm is mostly motionless. However, if you were to look at your arm with an infrared camera, it would be noticeably hotter than the rest of your body. Since heat radiates away, maintaining an elevated temperature requires work. So, your arm is performing work, but the work performed goes into heating your arm. No work goes into the weight because the weight is not changing: not in position, velocity, shape, nor temperature.

Answer 2

I find this topic to be one which confuses lots of people. The textbooks continue to use the "hold a book up" example, which causes quite a lot more confusion than I think it's worth.

Work is a force applied in the direction of movement over a distance. If there is no movement in the direction of the force, there is no work being done. If a table is holding up a heavy object, it may still be applying a very large force (exactly enough to counteract the force of gravity pulling down on the object), but it may do no work. This is a Good Thing(tm), because Work is also thought of as a change in kinetic energy. Energy is conserved, so if there was a way to gain energy while standing still, all sorts of perpetual motion machines would be possible!

However, when you bring the human body in, it gets more complicated. The human body is not as simple and static as a table. It is constantly adjusting and tweaking the balance between the muscle fibres to try to keep the object in your hand stationary. Too many muscle fibres engaged and you'd lift the object up. Too few, and it would fall. In theory, if you were to hold the sacromeres in the muscle fibers perfectly still at just the right moment, you would be able to hold up the object without expending any energy. In reality, this is just not how the human body works.

Thus, the human body is a very poor example for studying the difference between force and energy. It's simply trying to do too many other things at the same time.

In your heavy stone and wood table example, the direction of motion matters. In this case, everything is moving down (under gravity). The table, however, is pushing upwards on the heavy stone. Since the force is in the opposite direction of the motion, the work done is negative. Another way of saying this is that work is being done to the table, not the other way around.