Why don't atoms weigh anything?
Oh, but they do weigh something! Weighing sub-atomic particles is one of the things we do here at Jefferson Lab. You can imagine though that the scale we use to weigh particles is pretty bizarre. The weight of a particle is an important piece of information needed to figure out the properties of matter. Knowing the weight of atoms helps solve a completely different, but related, problem of how much the universe weighs. Believe it or not, there are scientists who stay up late at night trying to figure what the universe weighs.
You can think about how we weigh pieces of atoms with a sort of experiment in your mind. Imagine having a bucket of different balls that you could not see, but believed were there and you wanted to know how much they weighed. You also have a wall that the balls will leave a mark on if they hit it. Now place a fan blowing directly across the path between you and the wall. Throw the balls at the wall making sure you throw them all the same speed. The fan will push the lighter balls more to the side than the heavy ones. The lighter balls will hit the wall more to one side than the heavy balls. You could calibrate such a device to read weight by where the ball leaves the mark on the wall. That is sort of how we measure the weight of atoms. We shoot the particles through a magnetic field which pushes them to the side and then they go through an electronic detector that tells you where the particle passed.
There is another part of this question whose answer gets way out there. Scientists don't really know WHY things weigh anything at all. There are two kinds of particles, the kind that make up matter and the kind that transmit forces, like magnetism or gravity. Some people believe there might be a particle that is responsible for things having mass. Even though this particle has never been observed, it has been named already as the Higgs particle. Now think how much fun we could have if we learned how to control the particles that control weight.
Brian Kross, Chief Detector Engineer (Other answers by Brian Kross)