**4.4: Weight and Mass**

You probably remember from somewhere in high school science that weight and mass are different, but that there are lots of contexts where they at least SEEM to be interchangeable. Here, we’ll get a physicsy sense of the terms that should make the distinction clear.

Imagine you’re completely naive to science, but you have a scale and you step on it. The scale will say something like “you weigh 160 pounds.” What the scale is actually measuring is how much your body pushes down on a spring inside the scale. The spring is calibrated so that an object that pushes down with 160 pounds of force results in a dial turning to point at the number 160.

Now, suppose you took that same scale to the moon. The moon’s surface gravity is roughly 1/8th of Earth’s. So, if you were on the moon and stood on the scale, it would say 20 pounds. Importantly, it would say roughly 20 pounds no matter where you were on the moon.

So, we know that it’s measuring SOMETHING because it’s consistent for a given environment. But, we also know there’s something external which means that it’s different on the moon than on the Earth.

Those two things are (1) mass and (2) local acceleration of gravity

So, on Earth, you can say that the force on the scale is given by mg, where m is your mass and g is how much acceleration you experience due to gravity when you’re on the surface of the Earth. The unit of measurement for force in the metric system is the Newton, named after the short lived PDA put out by Apple in the 1990s.

Now, what is mass? You’ve probably been told something like “mass is how much stuff there is in a given space,” and that’s a pretty decent way to wrap your head around it. However, when we get to relativity, we’ll see the picture is a little different. I like to think about it like this: There is a certain class of things in the universe which alter space in such a way that other things tend to move toward them. The more such things there are, the more strongly space gets altered. The Earth, roughly speaking, has about 8x more such things than the moon. Thus, the things in your body have an 8x more powerful tendency into the Earth than they would on the moon. It also means that your body is slightly altering space to move Earth toward it. In fact, a question you might consider is that (to an outside observer), is the Earth moving you or are you moving the Earth?

Of course, for now, the math is much simpler. Any time you know the mass of an object (usually given in kilograms), and you know that it’s on Earth, you can multiply by g (~10 meters per second per second) to get the downward force it experiences in Newtons. And, if you know how many Newtons, just divide off g to get the mass.

Take note of the basic units that make up a Newton: kilograms*meters/seconds/seconds. It’s worth rolling them over in your head to make sure they make sense and don’t seem abstract. They measure the mass of something times how much it changes its velocity during a certain period of time. That right there is the essence of the universe. The Newton is a measurement of how much something in the universe is changing via an interaction with something else. Cool, right?

I know there’s a tendency not to get excited about simple things like this, when there are sexier topics like quantum mechanics, and circuits, and black holes, but these concepts of mass and weight are really cool by themselves. Understanding what they mean can change the way you look at everyday objects and distant galaxies.

**Next stop: Newton’s Third**

The Earth is actually about 80 times as massive as the moon. Things only weigh 6 times as much on the earth than the moon because it has a much larger radius than the moon – the surface is further away from most of the mass, so it has a lesser influence than it would if you scrunched the earth down to the size of the moon,

You described mass in terms of gravitation, but you can describe it equally well and possibly more clearly in terms of inertia. Say you have two cardboard boxes of equal size, one filled with printer paper and one filled with empty pop cans, and you want to slide them across the room. Which resists your push more? In other words, if you apply a similar amount of force, which one accelerates less? The one with more mass.

OK, so I ignored friction there, and friction does depend on gravity, though it shouldn’t outweigh the difference in mass. Imagine that you have these boxes on wheeled carts and also someone has spilled super-slippery oil on the floor.

Not convinced? Another approach requires either enough money to book a trip on one of those microgravity planes or the willingness to believe astronauts or passengers on those planes. Whether you go far enough to experience long-term microgravity, like on the Space Station, or just take a quick trip into microgravity thanks to the plane’s accelerating frame of reference, you experience next to no weight. If you took a shot put and a whiffle ball up there, they would also have very little weight, yet you would still find the shot put much harder to push than the whiffle ball thanks to inertial mass.

@Harold Tessmann III :

Correct me if I’m wrong, but I think that Gravitational and Inertial mass are not the same !

But they seem to be rigorously proportional, allowing us to set the Gravitational constant in order to have the same units and values for Gravitational and Inertial masses.

Would be fun though, to have different values : small weight & big inertia or the other way round