Monday, August 30, 2010

Laws and Definitions: Newton’s Laws of Motion

I find that in a lot of textbooks take definitions for granted and they do not differentiate laws and definitions. Let me be clear by what I mean by definitions. Most things cannot even be defined in the dictionary sense. For example one cannot really define what time is. But we can measure it. Quantitative relationships, which are the core of physics, come from quantitative measurements. We can take a periodic event as our standard measure of time and quantify time. We can also pick a unit of length and define a frame of reference in order to measure distance and position. We can then relate distance and time and find relationships such as centripetal acceleration to tangential velocity.

In order to demonstrate how most source do not properly define entities, I am going to discuss Newton's laws of motion and explore mass and force. Let's take Newton's second law: F = ma. Acceleration is a well-defined quantity which can be observed and measured. But what about mass and force? We might have some conceptual ideas about them but how do we measure them? Most resources say force is the thing that causes an object to accelerate. And mass or inertia is the resistance of the object to acceleration. Those are a good conceptual definitions, but this is mathematical truism. The measurement of force depends on mass and the measurement of mass depends on force. One quickly realizes that this is a useless relationship on its own because it does not tell us anything. When two quantities are undefined in a mathematical statement, its neither a law nor a definition - it's just useless. This is where most textbooks fall short because they do not define these quantities.

So how can we define and measure mass and force and give meaning to Newton's second law? It is clear that we need to define these values based on some measurable entity. Realizing this problem, the Austrian physicist Ernst Mach reformulated Newton's laws of motion based on measurable entities. He summarized Newton's laws into a single law:

"When two compact objects [point masses] act on each other, they accelerate in opposite directions, and the ratio of their accelerations is always the same. "

There is no mention of mass or force in this law; only acceleration. One can then quantitatively define mass and force based on this law. First we define an inertial frame of reference. Take the set of isolated point masses A and R, where R is takes as a reference point mass. At any time, if A is accelerating, R is accelerating in the opposite direction and the ratio of the accelerations is a constant. We define the mass of R as our reference mass with a unit of 1. Therefore, the mass of A is the ratio of the acceleration of A to R. Now mass is a measurable entity. Then we define force as mass times acceleration so that a unit of force (N) accelerates one unit of mass (kg) at 1 unit of of acceleration (m/s^2).

You have probably noticed that this definition is based on Newton's third law, which does not hold in relativity and thus in forces under Lorentz Transformation. Therefore, we need to modify and generalize our definition of mass. This definition is given in the paper "A Rigorous Definition of Mass in Special Relativity" written by E. Zanchini and A. Barletta. There is also a more formal version of the definition of mass in Newtonian mechanics. I highly recommend reading the paper:

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