# Understanding Newton's 2nd Law

• Vigorous
In summary, the conversation discusses the law of physics known as F=ma and how it can be measured and confirmed through various experiments and methods. It also delves into the relationship between force and mass and how they are defined in terms of each other. The conversation also addresses concerns about circular reasoning and explains the role of experimental verification in determining the function for F in the equation.
Vigorous
F=ma
To check if this law works, we measure the left hand side and the right hand side and if they are equal then the law is working.
To measure acceleration, we rapidly measure three positional measurements. Without appealing to the notion of the pull of gravity on an object, we measure the mass as follows, take a chunk of some material and define it as 1 Kg then proceed to find masses of other objects using a spring. We take a spring hook one end of the spring to a wall and displace the other by some amount x from equilibrium. Measure its acceleration. Then take the unknown mass and displace it by the same amount x. Assuming that under the same conditions the spring will exert the same force as in the case of 1Kg mass. So mE=1. a(m=1)/a(mE). From this know we know how any object can be attributed a mass and that mass is a measure of the resistance to acceleration for a given force. But now, how can we measure F without circular reasoning and saying that F=ma or when making a premise we are assuming its true?

Newton's second law introduces both force and mass. We assume we can calculate acceleration from position and time measurements, both essentially force and mass are defined in terms of each other.

There are ways to measure masses relative to each other. By using scales, for example, you can see when two masses are equal and also when the sum of two masses is equal to a third mass etc.

You can then do experiments using one mass falling under gravity and pulling a second mass along a table to verify the ##F = ma## relationship in that case.

There seem to be a few videos on line showing this if your search for "how to verify Newton's second law".

Vigorous said:
But now, how can we measure F without circular reasoning and saying that F=ma or when making a premise we are assuming its true?
Is it really circular reasoning? You may be worrying too much about this.
We can easily find that proportionality is followed. Distance and time are easy to define in terms of reproducible units (metre, c, second) so F can be shown to be proportional to m, whatever units we choose for them.

PeroK said:
Newton's second law introduces both force and mass. We assume we can calculate acceleration from position and time measurements, both essentially force and mass are defined in terms of each other.

There are ways to measure masses relative to each other. By using scales, for example, you can see when two masses are equal and also when the sum of two masses is equal to a third mass etc.

You can then do experiments using one mass falling under gravity and pulling a second mass along a table to verify the ##F = ma## relationship in that case.

There seem to be a few videos on line showing this if your search for "how to verify Newton's second law".

Use standard mass and balancing the unknown mass by adjusting the its position from the fulcrum point (idea of a weighing scale) is appealing mass to the notion of the pull of gravity there is no mention of that in F=ma

Vigorous said:
But now, how can we measure F without circular reasoning and saying that F=ma
With a spring.

nasu and russ_watters
Vigorous said:
Use standard mass and balancing the unknown mass by adjusting the its position from the fulcrum point (idea of a weighing scale) is appealing mass to the notion of the pull of gravity there is no mention of that in F=ma
There's a lot on line about how to confirm the second law. The law doesn't include details of the experimental aspects of verifying it!

You could look at how Arnold introduces the second law. Simply that for a system of ##n## particles described by positions ##x \in \mathbb{R}^{3n}## and velocities ##\dot{x} \in \mathbb{R}^{3n}##, that there exists a function ##F## such that ##\ddot{x} = F(x, \dot{x}, t)##, which along with ##6n## initial conditions uniqely defines the motion.

The actual function has to be determined experimentally. For example, for a set of uncharged particles in space you could look to Newton's gravitational force law to try and write down a form for ##F##, and then you have to hope that Newton's force law is a correct-enough model that the motions you solve for are a good fit to what you actually see in the real world.

Also, there are some restrictions on ##F##. For example form invariance under the full group of Galilean transformations ##G(R, v, a, s)## constrains that for inertial frames this function must only depend on the relative positions ##\{ x_i - x_j \}## and relative velocities ##\{ \dot{x}_i - \dot{x}_j \}## of all of the particles.

Vigorous said:
But now, how can we measure F without circular reasoning and saying that F=ma or when making a premise we are assuming its true?

There's often some ambiguity about which physical laws (e.g. Newton's 2nd and 3rd laws) should be considered as definitions and which are actual statements about physical reality. This is discussed, for instance, on page 50 of "Classical Dynamics of Particles and Systems, 5th ed." by Marion and Thornton. Anyway, the laws together contain enough information for you to determine by experiment whether they are consistent with real-world physics.

## 1. What is Newton's 2nd Law?

Newton's 2nd Law of Motion states that the acceleration of an object is directly proportional to the net force acting on the object and inversely proportional to its mass. In simpler terms, this means that the larger the force applied to an object, the greater its acceleration will be, and the more massive the object, the less it will accelerate.

## 2. How is Newton's 2nd Law expressed mathematically?

The mathematical expression of Newton's 2nd Law is F = ma, where F is the net force applied to an object, m is the mass of the object, and a is the resulting acceleration. This equation shows that force and acceleration are directly proportional, and mass and acceleration are inversely proportional.

## 3. How does Newton's 2nd Law relate to everyday life?

Newton's 2nd Law can be observed in many everyday situations. For example, when pushing a shopping cart, the harder you push (apply force), the faster it will accelerate. On the other hand, a heavier shopping cart will require more force to accelerate at the same rate as a lighter one. This law also explains why it is easier to ride a bicycle downhill (less force needed to accelerate) and harder to ride uphill (more force needed).

## 4. What is the difference between mass and weight in relation to Newton's 2nd Law?

Mass and weight are often used interchangeably, but they are not the same. Mass is a measure of the amount of matter in an object, while weight is a measure of the force of gravity acting on an object. In relation to Newton's 2nd Law, mass is a constant value that affects an object's acceleration, while weight can vary depending on the strength of gravity.

## 5. How does Newton's 2nd Law apply to objects in free fall?

Objects in free fall, such as a ball dropped from a height, are only affected by the force of gravity. According to Newton's 2nd Law, the acceleration of the falling object will be directly proportional to the force of gravity and inversely proportional to its mass. This means that all objects, regardless of their mass, will accelerate towards the ground at the same rate (9.8 m/s² on Earth).

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