Why do objects always 'tend' to move in straight lines? How come, everytime I see a curved path that an object takes, I can always say that the object tends to move in a straight line over 'small' distances, but as you take into account the curvature of the path, a force acting on the particle appears. I mean, I can always take a small enough portion of the curve, zoom in enough, and conclude that the object is moving in a straight line, but then as I zoom out I find out that a force is acting on the particle. The force of gravity is everywhere and, no matter how weak it is, it will make the particle take a path which is different from a straight line. This is my question: since particles are, in reality, never moving in straight lines, is Newton's first law a mathematical formalism or some true property of material objects?

# [Physics] Is Newton’s first law something real or a mathematical formalism

calculusinertial-framesnewtonian-mechanics

#### Related Solutions

To the first question: Newton's third law states that every force has an equal and opposite force. Thus, the force that Earth exerts on you with gravity (a big mass causing a huge acceleration on a small mass) is countered exactly by the normal force of the ground (again, a big mass causeing a huge acceleration on a small mass, but this time in the opposite direction). You can see this when you jump, and the normal force is no longer applied to your body, you quickly fall to Earth. In a sense, the force the ground pushes back on you is really equal and opposite to the gravity of the Earth pulling on you.

I'm not sure I understand your second question. It doesn't really specific the situation well, and it's poorly worded; if you rephrase it I can hopefully give you answer.

There are two aspects to this question. One is about geometry, and the other is about mechanics.

From a geometrical point of view, at every instant in time, *any* motion of a rigid body in 2D space that involves rotation is equivalent to a rotation about some *fixed* point in space. For example, at any instant a car wheel is rotating about the fixed point in contact with the road. Of course the "fixed" point is different at each instant in time, as the car moves.

On the other hand, when doing mechanics it may not be very interesting to know which point is "fixed," especially if the "fixed" point is not actually inside the object. Knowing the position of the "fixed" point is only important if there are some constraints on how the system can move, which apply some forces at the fixed point to *make* it a fixed point - in the car wheel example, the normal (weight) and tangential (friction) forces between the wheel and the road.

If there are no such forces, the simplest way to describe the motion in Newtonian mechanics is as a rotation about the center of mass, plus the translation of the center of mass. That is because the equations for the translation of the COM and the rotation about it are *independent* of each other, so they can be solved separately.

## Best Answer

Nice question! The answer to this depends on the version of Newton's first law you use.

In the Principia, the statement of the first law, as translated by Machin, is:

This is immediately followed by a series of examples:

Of the three examples, not one involves motion in a straight line! Since the first law is stated in the Principia in words rather than equations, there's a lot of room for ambiguity. Keep in mind as well that scientists reading the Principia in that era didn't know calculus, and vectors weren't invented until centuries later. Newton had to write in language his contemporaries would understand, even if it was at the cost of precision.

There are many different ways in which the first law has been stated over the years, as described in this question: History of interpretation of Newton's first law .

You can modify it to be a statement that if you choose a specific axis $x$, then the absence of any forces in the $x$ direction gives $dv_x/dt=0$ at that instant in time. This is probably the interpretation that's most directly suggested by Newton's three examples.

You can modify it to be a statement about objects that are acted on by zero

totalforce.As described in the other question, it's now popular (probably due to the influence of the analysis in Mach 1919) to describe it as a statement about the existence of inertial frames.

Gravity does present some unique issues, since it's a long-range force and can't be shielded against. Mach 1919 gave a very thorough and insightful critique of the logical basis of Newton's laws. Here is my own presentation of the question of what the first law really means and some experimental tests. In general relativity, we define a free-falling frame as an inertial frame, so that the motion of a projectile is defined to be "straight."

Ernst Mach, "The Science Of Mechanics," 1919, http://archive.org/details/scienceofmechani005860mbp