Let’s start off this section by doing a simple work calculation. Let’s say that you throw a ball up into the air.  The ball moves straight up a distance y,  and then falls back down the same distance, where you catch it.  We wish to find the total amount of work done on the ball by the gravitational force from the initial point (when the ball left your hand) until  the final point (when you were about to catch the ball). 

Recall that  work is just a component of force times a distance.  Taking the positive y-direction to point vertically upwards, we then have that, for the first part of the ball’s motion,

W_mg,1 = mg_y y = – mgy .

(Note that we have now taken the y-component of the weight force to find the work instead of the x-component. This is because the direction of the displacement of the ball is now called the y-direction.) Likewise, for the second part of its motion as the ball is falling back downwards, now calling the positive y-direction downwards (since  the displacement is downwards – it’s easiest to always call the direction of the object’s displacement the positive direction when computing work), we get that

W_{mg, 2} = mg_y y = + mgy .

Therefore, with the ball’s motion starting and stopping at the same position (your hand), we find that the net work done on the ball by the gravitational force is

W_mg = W_{mg, 1} + W_{mg, 2} = 0 .

The gravitational force (weight) does zero net work when the starting and ending points are the same.  We say that such a force is a conservative force.  (With a bit of reasoning, we can also see that a conservative force is a force whose work is independent of path – that is, it doesn’t matter how you go from the initial to the final point, the work done by the force is still the same.)  This means that, during part of the motion the force is doing positive work (meaning that the force is transferring energy to the object), and during part of the motion the force is doing negative work (taking energy awayfrom the object). Whenever the force is helping the object move in the direction it is moving, the work done is positive.   On the other hand, whenever the force is trying to stop the object from moving in the direction it is moving, the force is doing negative work.

There are twoconservative forces with which we are presently familiar: the gravitational force and the spring force.  The one force that we know of that is nonconservative is the friction force (this force always does negative work – that is, it always opposes the motion of the object!).