# Why does exerting force sometimes cost energy and sometimes not?

Inspired by these similar questions, which have some very good answer to what they are, but they all focus on what is happening. None of them explains why it only applies sometimes and not others.

I'm looking for a more general explanation, not specific to a single case. There are many situations where energy has to be expended to hold something up - a hovering helicopter, lifting with your arm, using an air jet to simulate skydiving - and so on. In all of these cases, an object is being held in a static position (like a book on a table), but energy has to constantly be expended to maintain it (unlike the book). Why is this?

A hovering helicopter is falling. In order to maintain it's position, it must produce a downdraft - spending energy to force air downward to lift itself up. It is exerting a force on the air. A book on the floor requires energy to be spent to lift it to the table. In both cases the same amount of energy is expended: An amount equal to the amount of potential energy gained by the object. The difference is, once the book has been moved, it stays there. No more energy is being spent by it or the table to hold it up, despite the fact that the book is exerting a force on the table - just like the helicopter is on the air. Except that the helicopter has to keep spending energy to exert that force, and the book doesn't.

Energy is being transferred from the helicopter (chemical, in the fuel) into the air (as motion). The helicopter is, net, losing energy in order to maintain it's position.

Why can force sometimes be exerted for free (the table has a constant force on the book), but sometimes has a cost?

• The helicopter has to continue to "do work on the air" to keep itself up. The book does not have to do work to stay on the table. If the air was dense enough then the helicopter would not need to do anything to stay up. – Aaron Stevens Aug 29 '18 at 14:58
• @AaronStevens, right, but why the difference? Why does the helicopter need to do work, when the book doesn't? – Benubird Aug 29 '18 at 15:02
• Because the book is held up by the repulsion force of the table. In order for the helicopter to have enough of a repulsion force from the air it needs to rotate its propellers to allow enough air molecules to be close enough to the blades. – Aaron Stevens Aug 29 '18 at 15:05

There is a change in energy if a force is exerted on a thing that is moving, as a formula $\Delta E = W = \int \vec F \cdot \text{d}\vec{s}$ where the integral is calculated along the path of the moving thing. If nothing moves, the change in energy (the "cost") is zero. In your example, the air is moving, and the helicopter exerts a force on it.
In order for the book or the helicopter to stay at rest, it must be that the net force acting on each object is $0$. This means the force holding the object up has to be equal to its weight.