Newton’s second law of motion establishes a relation between the strength of the force acting on s body and the body’s acceleration. But the definition can do more than your subjective feeling of when a force is weaker and when it is stronger: It allows you to determine the ration of force strengths – we can measure when a force is two times, three times, four times as strong as another. The definition meshes with our everyday ideas of when forces are strong: pull more strongly, and the spring will be stretched further than by a weaker pull. As millions of high-school students have witnessed, one can measure the strength of a force using suitably chosen springs: The elongation of a spring is a measure for the strength of the force acting on it – at least in cases where the spring’s deformation stays within certain limits: The mass we have defined in this way plays a key role in classical mechanics – that is, in the laws that describe how objects move under the influence of forces. While the collision hardly affects the more massive ball – a little slowing down, that’s all – the cyan ball’s speed changes quite drastically: it bounces and moves back into the direction it came from, and at significantly higher speed than before. However, the mass of the cyan ball is just a quarter that of the violet one. The animation above shows an example: Before the collision, the cyan and ball and the violet one are moving towards each other equally fast, but in opposite directions. In consequence, the collision is going to cause the speed of the first body by a comparatively small amount, compared with the change in the second body’s speed. If, say, the first body’s mass is much greater than the second body’s, then the left-hand side of the above equation is a large number. This definition is consistent with our everyday notion of larger and smaller masses (or larger and smaller inertia, respectively). As soon as we have defined the unit of mass (in other words, chosen one particular body as a standard), we can make experimental collisions and thus determine the masses of all other bodies. Where the minus sign is necessary to obtain a positive ratio. We can use this fact to define the masses of the two objects, using the relation In other words: this ratio depends only on the object’s themselves, not on their motion. If you make this experiment, you will find that the ratio between those two changes in speed is independent of the object’s initial speeds. Let’s call the first object’s change in speed Δv 1, and the second one’s Δv 2, where each time, the change is defined as the object’s speed before the collision minus its speed afterwards. In general, the collision will change the speeds of the spheres. These objects move along a single line before and after the collision: The easiest case is a head-on collision between two spherically symmetric objects. Your typical high-school physics laboratory probably has an air track on which gliders will move almost without friction, which will do the trick. In the laboratory, one would first need to eliminate friction. In outer space, it would be sufficient to have these objects float towards each other. Instead, it is more useful to look at simple collisions between different bodies. While this example gives a rough idea of what inertial mass is all about, it doesn’t lead directly to a definition: to quantify mass in this way, we would need to define when two shoves have the same strength. The difference between ball and box? The box’s significantly greater mass, which is responsible for the box reacting so much more diffidently than the ball to the same amount of shoving. It does drift away from me, but only very slowly. The ball reacts to my shove by accelerating and flying away at high speed.
Now I give the ball a shove and, afterwards, the box, taking care to exert exactly the same strength both time. Next to me, a box and a small ball are floating in space. I’m standind on the outside of a space station, safely held in place by clamps fixing me to the station’s hull. Imagine that I’m in empty space, properly suited up, far away from all major sources of gravity. This difference, in turn, was Einstein’s starting point in developing his theory of gravity, space and time: general relativity. In classical physics, mass plays a curious double role, which is responsible for a peculiar difference between gravitation and all other forces.
General relativity and the weak equivalence principle.