# Understanding Newton's 2nd Law

• Vigorous

#### 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".

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?
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.

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

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

• nasu and russ_watters
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.

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.