# Confusion about the relation between Newton's first law and second law [closed]

I just started learning Physics in college and it seems like there are so many versions of the Newton's laws of motion and it also seems like I should go with belowe as the Newton's first law and second law (These are my my interpretation after reading a bunch of things online).

Newton's first law: For any object, there exists an inertial frame of reference. An inertial frame of reference of an object is one such that its acceleration is the zero vector if and only if the net force on it is the zero vector.

Newton's second law: In any inertial frame of reference for an object, there exists a constant, called mass, such that the vector sum of the forces on it always equal to the scalar multiplication of its mass and acceleration.

First of all, it seems like the term "force" in the Newton's laws, like "acceleration", already have definitions and measurements, so the Newton's laws are based upon those definitions and are laws about those quantities. Acceleration means the second derivative of the displacement vector of an object which is measured with respect to a particular frame of reference. And a force is just a push or pull, already measured by something else before the Newton's laws were put forward? With these in mind, it seems like Newton's second law is a definition of mass? Is there a concept of mass before the Newton's laws were put forward?

Furthermore, it can be seen the Newton's first law above is true because the object itself is an inertial frame of reference of itself; its displacement is always zero. It can also be seen in certain reference frames, the second law fails, one example being an ice puck being sent down the north pole in a long path with a point on ground as the reference point.

After reading the first two laws, it seems like the first law is supposed to set up a reference frame in which the second law holds. However, it is hard for me to see the connection between the two laws. Also, considering a ball that is in free fall, the net force on it is not zero, but its acceleration with respect to an inertial frame of reference (itself is zero). So, I cannot see any connnection between the reference frame for which the second law holds, and. the reference frame set by the first law. What is the connection?

• I don't know what your sources are, but I see a lot of confusion in your statements. I suggest you read the following for a coherent explanation of all three Newton's laws of motion: hyperphysics.phy-astr.gsu.edu/hbase/Newt.html#nt1 Commented Mar 1, 2022 at 21:54
• it seems like there are so many versions of the Newton's laws of motion No, there is not. Sometimes they are explained in different ways, but they are all fundamental laws and are unique. As stated by @BobD there are many misconceptions here, and it might help you if you studied up on the subject first. Commented Mar 1, 2022 at 22:16
• @josephh I can find online laws that involve refrence frames and ones that do not. There are ones that claim to be modern versions.
– TFR
Commented Mar 2, 2022 at 1:13

1. The first law does not say that there is an inertial frame of reference for any object. In fact this statement doesn't make much sense. The first law says that in an inertial frame of reference the acceleration is zero if and only if the force is zero. An inertial frame of reference is a frame that is not accelerating. You can tell if you are accelerating or not, acceleration is what glues you to the seat of your car when you are overtaking a car on the highway, so in principle you can always say if the frame you have chosen is an approximately inertial frame of reference or not.

2. Newton's second law could be made to work (kind of) in a non inertial frame of reference but it's tricky (one has to add a pseudoforce) and I don't want to add to your confusion so I'll just say that the same goes for the second law: $$\textbf{F} = m \textbf{a}$$ only holds in an inertial frame of reference. Experimentally you have to measure the force. For example, you could work out experimentally. For example you could drop objects with a different mass (the mass can be defined in units of some other weight by using a two-armed scale), measure the time it takes for them to fall, repeat the experiment dropping them from different heights many times, trust Newton's second law as an axiom, and finally by extrapolating the data you should get something like $$F_z = -mg$$ with the other components being zero. Once you have built a model for your force you can use it to calculate stuff for all kinds of other objects assuming that $$F_z = -mg$$ holds for them as well.

3. The third law holds for all reference frames in classical mechanics (electromagnetism complicates matters a bit here though, because the electromagnetic field carries momentum and in order to have Newton's third law hold you should account for that momentum as well, and at first sight this is not exactly intuitive).

• What do you mean "an inertial frame of reference is a frame that is not accelerating"? Isn't acceleration always measured with respect to a reference frame? With respect to which reference frame is the reference frame you mentioned not accelerating?
– TFR
Commented Mar 2, 2022 at 1:11
• Your answer adds more confusion than resolution. Newton's laws are not made with frames of reference in mind. It is wrong to involve relativistic effects with Newton. Commented Mar 2, 2022 at 7:17
• They might not be made with them in mind but still, the first law holds only in an inertial frame of reference. I guess one could state the first law and then define inertial frames as those frames where the first law holds, or something along these lines. Commented Mar 2, 2022 at 21:10

You might find it interesting to read Newton's original statement of his laws in his 'Principia' which is readily available online in English (he wrote it in Latin).

His first law just says that if a body is left undisturbed its state of motion won't change, so if the body is stationary it will remain stationary and if it is moving at a constant speed in some direction then it will continue moving with that speed in that direction.

His second law, in its original form, says that when you apply a force to a body the body will accelerate in the direction of the applied force, and the value of the acceleration will be proportional to the magnitude of the force, so if you double the magnitude of the force you will double the acceleration.

The concepts of force, mass and acceleration all pre-date Newton, but there were conflicting ideas of how they related to each other. What Newton did was to formulate the rule that the force and the resulting acceleration were proportional to each other for a given mass.

Frames of reference are a more modern idea (you won't find any reference to the phrase in Newton's book). You also need to bear in mind that there are two entirely separate conventional ways of talking about forces and inertial motion, namely the Newtonian approach, in which gravity is treated as a force, and the ideas that have arisen from General Relativity, in which gravity is not a force but just a consequence of the curvature of the geometry of the spacetime through which objects move.

So in everyday Newtonian language (which is the language used to teach physics without reference to general relativity), an object in outer space, where there is negligible gravity, will move inertially in the absence of forces. Elsewhere, gravity acts as a force that will cause the object to accelerate (ie to move non-inertially in Newtonian-speak).