Why is work defined as force times displacement?

[aftershock]

I know how to use the formulas and solve problems, but fundamentally what exactly is work?

Why is force times displacement a useful quantity? How do we know that in order to do work we need energy, and that energy must be conserved?

 

[chingel]

Almost all of the everyday interactions are based on electromagnetic and gravitational forces. These all get weaken the further away you move from the source by the inverse square law. If you have one charged particle next to another of the same charge, they will repel. This force will depend on the distance. If you let the particle move by a distance away from the other one, the force on it will now be less and it will not go back to the first position unless you apply exactly the same force through the same distance on it as it felt when moving away.

Basically there is always a symmetry, the force it is applied to in one direction is always the same as the force you need to apply in the other direction to get it moving the other way. When you let the particle move further away, it feels less force and it will not feel as strong of a force from the other particle again unless you move it back by applying the same force that was applied to it. So by moving (along the gradient of the force) you let some of the force vanish, which will not come back, unless you force it.

So for example you want the wheels of the car to rotate and you burn some gasoline. It's probably an oversimplification, but you break up the gasoline with oxygen and then you have atoms or molecules very close to each other that repel each other by electromagnetic forces. These forces accelerate the molecules, hit the piston, push it because of inertia and the piston turns the wheel. The electromagnetic repulsion had to be overcome beforehand to put the molecule together and that is exactly how much force that came out of it.

This also means that when an object is moving at a right angle to the force gradient, the force it could feel moving away from it stays the same (i.e. the potential to do work moving along the force gradient is the same) and that is why we say that no work is done when one object is orbiting another.

 

[rbj]

Why is force times displacement a useful quantity?

think about what you learned about levers in elementary/middle school. you could lift a 1 lb object up 1 foot by directly lifting it. or you can lift that same 1 lb object up that same 1 foot distance using a lever with a moment arm that is twice as long for your lifting hand as it is for the object. your lifting hand would need to apply 1/2 lb force, but you would have to lift the lever twice as far (2 feet).

but the net effect is the same, you got your 1 lb object lifted up by 1 foot. whether you had to exert 1 lb and 1 foot, or 1/2 pound and 2 feet, or 1/3 pound and 3 feet, the net accomplishment is the same.

How do we know that in order to do work we need energy, and that energy must be conserved?

well, in my opinion, that became sort of an axiom. perhaps someone can construct an argument, strictly from Newtonian mechanics for linear and rotational motion, why it must be true. maybe something like: if work, defined as force x displacement, is not a conserved quantity, then one can construct a situation where a mass lifted up 1 meter could be lifted up another meter without any "effort" coming in from the outside. it would mean things moving against the gravitational force without any other force acting on it.