# Why is force invariant in Newtonian mechanics?

• Amin2014
In summary, the guy on the ground doesn't understand how gravity works in a car, because he is not in the car and cannot measure the forces. He concludes that there must be gravity because the car and all the objects inside it follow a horizontal trajectory.
Amin2014
Recently, I've been pondering deeply on relativity (both Galilean and SR) and all of a sudden I find that I don't grasp even the basic concepts of physics (or life) anymore, i.e. I can't go back to my previous, "normal" mode of thinking.

Consider Newtonian mechanics, take the ground to be at rest. Imagine a car moving with constant velocity relative to the ground. We have two observers, one standing on the ground and one inside the car. A bob is hanging from a string attached to the ceiling of the car, making an angle alpha with the vertical line (the guy inside the car is applying a horizontal force P to the bob). We want to compare the formulation of Newton's second law for the two observers (and hopefully conclude that both observers find the law in the form of F=ma to be valid).
Now I know this is supposed to be a simple, elementary question, but like I said, I suddenly became LOST. How does the observer on the ground know what forces to put in the equation? The forces are "inside the car", how does he know what forces are being applied to the ball? How does he know their magnitude? He's not even in the car, so how is he measuring/feeling these forces? How does he measure the magnitude of g in the car from his own point of view? (recalling that g was 9.81 m/s2 when he was in his own car doesn't count)

How does the guy on the ground even know there's gravity in the car? How does he conclude this? Considering he's not inside the car, does he conclude this based on his past experience being in moving cars? No that can't be right, because it involves changing reference frames...does he conclude this based on Newton's second law? That can't be right either since it would be circular reasoning. Does he recognize gravity by considering the behavior of the "whole picture" (the car together with all the objects inside)?

How does he know that the bob has whatever mass it has? I mean how does he know what mass to substitute for m in the equation F=ma to check the equation's validity?

After some long thinking and mind twisting, I've finally come up with an answer to the above questions that I might share in my replies, but I would like to know what you guys think first. See attachment for pic.

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Amin2014 said:
How does the guy on the ground even know there's gravity in the car? How does he conclude this? Considering he's not inside the car, does he conclude this based on his past experience being in moving cars? No that can't be right, because it involves changing reference frames...does he conclude this based on Newton's second law? That can't be right either since it would be circular reasoning. Does he recognize gravity by considering the behavior of the "whole picture" (the car together with all the objects inside)?
Well, my answer will be probably trivial...but if the car follows a horizontal trajectory (respect to the ground system), there is no way to eliminate (or modify) the gravitational force (that is orthogonal) in the moving frame. Hence gravity must be present in both the reference frames.

Anyway, if you just consider two references (in non relativistic case) and you want to check Newton's law, I think it is sufficient to verify that in both system an object, put in a certain position, with no interactions, etc. remains in its own position (i.e. the two references are inertial). Then, the invariance (in form) of Newton's law should directly come from Galileian relativity.
In the special case of gravity: if you want to now that in a certain system there is or not gravity (or an equivalent acceleration), just leave a ball in the air (without any force applied) and if it falls down, there is gravity (or you are in a system that simulates the gravity)...

Amin2014 said:
He's not even in the car, so how is he measuring/feeling these forces?
Assuming the driver uses some system where you can read of the force, the observer outside can read of the value as well (in theory...).
Amin2014 said:
How does he measure the magnitude of g in the car from his own point of view?
Ask the driver to let something fall down, measure how fast it accelerates downwards.

And so on.

All those measurements might be impractical with a car, but they are all possible in principle, and even easy to do if you replace the car by some sled in a laboratory, for example.

Amin2014
The guy on the ground simply sees the ball moving at costant velocity, so his equation is simple: F=0. Than he can ask himself a simple question: if there's gravity in that car, why isn't it "acting" on the ball? Why can't I see any force? It is balanced by other forces. If there's no acceleration the sum of all forces is zero and measuring acceleration is simple becouse it is the variation of velocity which is the same for every observer (this can be easily shown in both galilean and special relativity)

Amin2014 said:
Considering he's not inside the car, does he conclude this based on his past experience being in moving cars? No that can't be right, because it involves changing reference frames...does he conclude this based on Newton's second law? That can't be right either since it would be circular reasoning.

An observer without any knowledge of physics would definitely not be able to say anything about forces even being inside the car.

An observer knowing some physics would simply assume that known physics holds inside the car, and interpret observations accordingly.

mfb said:
Assuming the driver uses some system where you can read of the force, the observer outside can read of the value as well (in theory...).
Ask the driver to let something fall down, measure how fast it accelerates downwards.

And so on.

All those measurements might be impractical with a car, but they are all possible in principle, and even easy to do if you replace the car by some sled in a laboratory, for example.
Great answer, I came to the same conclusion myself: When we say force or any other quantity is invariant between inertial frames, we mean that if the person inside the frame where the force exists applies some method to measure it, whatever number he reads, the observer on the ground will read the exact same number. In most problems in physics, we are GIVEN the mass or force and we take it for granted between the two frames, and we might combine this with circular reasoning to conclude that force or mass is invariant. But that's not what invariance is all about.

Let me explain: The guy inside the car can't "feel" the tension in the string. It's being applied to the bob, so how does he know what to substitute for T in his equation? Well, fist of all he sees that the string makes an angle alpha with the vertical, so he knows the direction of T. In order to find its magnitude, he needs to convert the tension to some length; that's how all physical quantities are ultimately measured. So he attaches a spring to the bob to replace the string (or the bob could've been attached to a spring in the first place), and he measures the change in length of the calibrated spring to find the magnitude of the force T that is being applied to the bob (this is the force needed to hold the bob in place by replacing the string with a spring). Now, the guy on the ground would observe the same angle for T, therefore concluding the same direction for T, and he would read the same number from the ground (he has to have strong eyes or a magnifier to read this number). So the magnitude that he measures indirectly for T would be the same as well. This is the gist of invariance in Galilean relativity.

If we want to measure any other quantity, we have to relate it to length. Even force P which is applied by the observer inside the car requires a calibrated spring for accurate measurement (prior to substituting it in the F= ma equation). If you want to measure temperature, you can't just touch an object and say "wow, that's hot!". That would be qualitative measurement. In order to make a quantitative measurement, you'll have to somehow convert or relate temperature to some quantity that you can see, which would be length. For length is something you can see, and thus measure directly. You might relate temperature to volume of mercury and read it from there. By doing so you are relating it to the length of mercury in the tube. You need to measure the mass of the bob so you can substitute it for m in your equation, and you could do this by using a mass balance (before hanging the bob of course). If you want to measure g, you'll have to relate it to some length by means of an experiment or instrument. Since measuring any quantity like this involves measurement of length, and this length is being measured at some instant in time, therefore essentially all the measurements of different quantities boil down to measuring some distance between two points at some frozen instant in time. Since we don't have length contraction in Galilean relativity, all such quantities will be invariant from one coordinate to another.

Only abstract quantities like position, velocity and acceleration are related to the coordinate system. Such quantities are not a measure of distance between two fixed points, but rather a measure of distance between one fixed point and the origin. Therefore they are generally subject to change by translation of coordinates. (Of course acceleration won't change if both frames are inertial)

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voko said:
An observer without any knowledge of physics would definitely not be able to say anything about forces even being inside the car.

An observer knowing some physics would simply assume that known physics holds inside the car, and interpret observations accordingly.
An observer on the ground doesn't KNOW beforehand what the mass of the bob is, or what the forces acting on it are, or what g is (it could be zero for all he knows- at best he may suspect gravity if he intuitively considers the "whole picture" of the car). The observer in the car doesn't know the magnitude of these quantities either, so he conducts experiments to find g and makes measurements to find the forces or mass of bob. He will read numbers off scales for each of these measurements. The observer on the ground, while observing the car, will read the same numbers during these measurements. Thus they will substitute the same quantities for force, mass or g. Up to here, neither observer knows what the other observer is measuring. They don't know or have access to the other observer's numbers. Once the two observers meet, they can compare their numbers to conclude that they've both read the same values.

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Amin2014 said:
An observer on the ground doesn't KNOW beforehand what the mass of the bob is, or what the forces acting on it are, or what g is (it could be zero for all he knows- at best he may suspect gravity if he intuitively considers the "whole picture" of the car).

This is not about intuition. This is about physical modelling. If the car is on the Earth, a competent observer can obtain a very accurate estimate of gravity within the car without measuring anything in the car. Other forces acting on the bob will not be known, nor will be its mass. But from the mere non-vertical position of the bob the ground-based observer can infer there are some non-zero forces different from gravity acting on the bob.

voko said:
This is not about intuition. This is about physical modelling. If the car is on the Earth, a competent observer can obtain a very accurate estimate of gravity within the car without measuring anything in the car. Other forces acting on the bob will not be known, nor will be its mass.
How can he obtain an accurate estimate of anything without making any measurements?
voko said:
But from the mere non-vertical position of the bob the ground-based observer can infer there are some non-zero forces different from gravity acting on the bob.
Inference without measurement, that's what I meant by intuition.

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Amin2014 said:
How can he obtain an accurate estimate of anything without making any measurements?

I did not say "any measurements". I said "anything in the car". That is different.

Amin2014 said:
Inference without measurement, that's what I meant by intuition.

I do not want this to degenerate into a discussion of semantics, but "intuition" usually means something not necessarily scientific. I, however, described how the scientific approach works.

voko said:
I did not say "any measurements". I said "anything in the car". That is different.
If not in the car, where else? What sort of measurement do you have in mind? Can you describe the steps by which the observer on the ground can obtain an estimate of g?

Amin2014 said:
How does the guy on the ground even know there's gravity in the car?

Because the known laws of gravity state
nothing about shielding from gravity by a car chassis.

Amin2014 said:
If not in the car, where else? What sort of measurement do you have in mind? Can you describe the steps by which the observer on the ground can obtain an estimate of g?

The gravitational field of the Earth is very precisely known. The effect of the car on the field can safely be neglected (unless the observer meticulously takes account of a myriad other small disturbances).

## 1. Why is force considered invariant in Newtonian mechanics?

Force is considered invariant in Newtonian mechanics because it is based on Newton's first law of motion, also known as the law of inertia. This law states that an object will remain at rest or in uniform motion unless acted upon by a net external force. In other words, the amount of force required to change an object's motion is dependent on the object's mass and acceleration, rather than its position or velocity. This principle holds true in all inertial reference frames, making force an invariant quantity.

## 2. What is the relationship between force and inertia in Newtonian mechanics?

In Newtonian mechanics, force and inertia are directly related. Inertia is the tendency of an object to resist changes in its motion, while force is the external influence that causes changes in motion. The more mass an object has, the more inertia it has and the more force is required to change its motion. This relationship is described by Newton's second law of motion, which states that the force applied to an object is directly proportional to its mass and acceleration.

## 3. How does Newtonian mechanics explain the concept of force?

Newtonian mechanics explains force as a vector quantity that is necessary to change the motion of an object. It is described as a push or pull on an object and is measured in units of newtons (N). According to Newton's third law of motion, for every action, there is an equal and opposite reaction. This means that when a force is applied to an object, the object will exert an equal and opposite force back onto the source of the original force.

## 4. Are there any limitations to the concept of force in Newtonian mechanics?

While Newtonian mechanics provides a useful framework for understanding the behavior of objects in motion, it is limited in certain scenarios. For example, this theory does not account for the effects of extreme speeds or the behavior of subatomic particles. Additionally, it does not explain the force of gravity, which is addressed in Einstein's theory of general relativity. However, for everyday objects and motions, Newtonian mechanics remains a valid and widely used theory.

## 5. How does the concept of force in Newtonian mechanics differ from that in other theories of physics?

The concept of force in Newtonian mechanics differs from that in other theories of physics, such as quantum mechanics and relativity, in several ways. In quantum mechanics, forces are described as interactions between particles, while in relativity, forces are seen as a result of the curvature of spacetime. Additionally, the laws of motion in Newtonian mechanics are deterministic, meaning that they can predict the exact future state of a system, while in quantum mechanics, there is an inherent uncertainty in the behavior of particles. However, all of these theories ultimately seek to explain the fundamental nature of forces and their effects on the physical world.

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