Is the principal of superposition of forces a part of the second law?

[Jiman]

The three laws are basic principles in Newtonian Mechanics. The principle of superposition of forces is part of the second law.But before we make the assumption of superposition of forces,we have to make another assumption which is the principal of superposition of motion.why are displacement，Velocity and acceleration all VECTORS？

 

[Lnewqban]

Good question!
In order to complete calculations, it is convenient to assign both, magnitude and direction to those.

 

[ZapperZ]

The three laws are basic principles in Newtonian Mechanics. The principle of superposition of forces is part of the second law.But before we make the assumption of superposition of forces,we have to make another assumption which is the principal of superposition of motion.why are displacement，Velocity and acceleration all VECTORS？

This is puzzling.

The requirement for something to be a vector is NOT that there has to be superposition. Superposition of vectors is nothing more than vector addition. I can do similar thing to, say, displacement and velocity (boat pointing in one direction, river flowing in another).

Vector addition is a consequence of something being a vector, not the other way around.

Zz.

 

[Jiman]

This is puzzling.

The requirement for something to be a vector is NOT that there has to be superposition. Superposition of vectors is nothing more than vector addition. I can do similar thing to, say, displacement and velocity (boat pointing in one direction, river flowing in another).

Vector addition is a consequence of something being a vector, not the other way around.

Zz.

The force can add Vectorily due to the principle of superposition of forces. But accereration and velocity can also add Vectorily due to which assumption??

 

[ZapperZ]

The force can add Vectorily due to the principle of superposition of forces. But accereration and velocity can also add Vectorily due to which assumption??

Once again, this is rather puzzling.

If F = ma, when if F is a result of the addition of forces from various direction, why can't acceleration also be the result of the addition of acceleration of various acceleration?

There are no "assumptions". It is the mathematics. Once you declare something as a vector, then all the mathematical rules that apply to a vector comes into play.

Zz.

 

[Jiman]

Once again, this is rather puzzling.

If F = ma, when if F is a result of the addition of forces from various direction, why can't acceleration also be the result of the addition of acceleration of various acceleration?

There are no "assumptions". It is the mathematics. Once you declare something as a vector, then all the mathematical rules that apply to a vector comes into play.

Zz.

The force can add Vectorily due to physical assumption!. Displacement,Velocity and Accereration can add Vectorily due to Mathematical Method,right?

 

[ZapperZ]

The force can add Vectorily due to physical assumption!.

What physical assumption?

Zz.

 

[Jiman]

What physical assumption?

Zz.

Force Displacement,Velocity and Accereration can add Vectorily only due to Mathematical Method,right?

 

[Jiman]

What physical assumption?

Zz.

The principal of superposition of forces is based on the mathematical method,right?

 

[vanhees71]

The three laws are basic principles in Newtonian Mechanics. The principle of superposition of forces is part of the second law.But before we make the assumption of superposition of forces,we have to make another assumption which is the principal of superposition of motion.why are displacement，Velocity and acceleration all VECTORS？

The upshot is. To start with physics (both experimental and theoretical) you need a spacetime model, and the Galilei-Newton spacetime model simply uses a bundle of 3D Euclidean spaces along a directed 1D time space. Taken together with Lex 1 (the special principle of relativity) that's the abstract formulation of Newton's "absolute space and absolute time".

 

[sophiecentaur]

The requirement for something to be a vector is NOT that there has to be superposition.

Superposition can only apply when you have an isotropic, linear medium. We assume that in most of our dealings with SpaceTime. Newton certainly did.

 

[Jiman]

The upshot is. To start with physics (both experimental and theoretical) you need a spacetime model, and the Galilei-Newton spacetime model simply uses a bundle of 3D Euclidean spaces along a directed 1D time space. Taken together with Lex 1 (the special principle of relativity) that's the abstract formulation of Newton's "absolute space and absolute time".

Thank you,my friend!

 

[Jiman]

Once again, this is rather puzzling.

If F = ma, when if F is a result of the addition of forces from various direction, why can't acceleration also be the result of the addition of acceleration of various acceleration?

There are no "assumptions". It is the mathematics. Once you declare something as a vector, then all the mathematical rules that apply to a vector comes into play.

Zz.

Thank you ,my friend!Are you student?Which university?

 

[Nugatory]

But acceleration and velocity can also add Vectorily due to which assumption??

If we assume that velocities are vectors, then it follows from the definition of acceleration (first derivative of velocity) that acceleration is also a vector. So your question comes down to asking why we can start with the assumption that velocities are vectors.

We do this because it leads to a useful mathematical treatment of real-world situations. For example: if two cars moving with velocities $\vec{v_1}$ and $\vec{v_2}$ collide we can subtract the velocity vectors to find the speed of the collision; we calculate the velocity across the surface of the Earth of a boat moving with velocity $\vec{v_1}$ across a river when the water is moving with $\vec{v_2}$ as velocity $\vec{v_1}+\vec{v_2}$

If vector addition didn't accurately describe how velocities work in the real world we wouldn't use vectors to describe velocities. We'd use some other mathematical formalism that did match the real-world behavior.

 

[jtbell]

For what it's worth, the vector addition of forces, in its geometrical interpretation, is given by Newton as "Corollary 1" to "Law 2" in his Principia Mathematica. In an English translation of Newton's Latin:

A body acted on by forces acting jointly describes the diagonal of a parallelogram in the same time in which it would describe the sides if the forces were acting separately.

This is accompanied by a parallogram diagram which is very similar to the one that you see in most (or all?) introductory physics textbooks today. It lacks only the arrows that normally represent vectors nowadays.

 

[Jiman]

If we assume that velocities are vectors, then it follows from the definition of acceleration (first derivative of velocity) that acceleration is also a vector. So your question comes down to asking why we can start with the assumption that velocities are vectors.

We do this because it leads to a useful mathematical treatment of real-world situations. For example: if two cars moving with velocities $\vec{v_1}$ and $\vec{v_2}$ collide we can subtract the velocity vectors to find the speed of the collision; we calculate the velocity across the surface of the Earth of a boat moving with velocity $\vec{v_1}$ across a river when the water is moving with $\vec{v_2}$ as velocity $\vec{v_1}+\vec{v_2}$

If vector addition didn't accurately describe how velocities work in the real world we wouldn't use vectors to describe velocities. We'd use some other mathematical formalism that did match the real-world behavior.

It is only mathematical method,not physical assumption,right?

 

[Jiman]

If we assume that velocities are vectors, then it follows from the definition of acceleration (first derivative of velocity) that acceleration is also a vector. So your question comes down to asking why we can start with the assumption that velocities are vectors.

We do this because it leads to a useful mathematical treatment of real-world situations. For example: if two cars moving with velocities $\vec{v_1}$ and $\vec{v_2}$ collide we can subtract the velocity vectors to find the speed of the collision; we calculate the velocity across the surface of the Earth of a boat moving with velocity $\vec{v_1}$ across a river when the water is moving with $\vec{v_2}$ as velocity $\vec{v_1}+\vec{v_2}$

If vector addition didn't accurately describe how velocities work in the real world we wouldn't use vectors to describe velocities. We'd use some other mathematical formalism that did match the real-world behavior.

Newton's parallelogram law is only mathematical method without physical significants,right?

 

[Jiman]

This is puzzling.

The requirement for something to be a vector is NOT that there has to be superposition. Superposition of vectors is nothing more than vector addition. I can do similar thing to, say, displacement and velocity (boat pointing in one direction, river flowing in another).

Vector addition is a consequence of something being a vector, not the other way around.

Zz.

Newton's parallelogram law is only mathematical method without physical significants,right?

 

[vanhees71]

It is only mathematical method,not physical assumption,right?

Well, the choice of a mathematical spacetime model is an assumption about how the phenomena may be best described. You need it to start with making measurements. E.g., to be able to measure the distance between two physical points you need a mathematical model about the geometry of space. In Newtonian mechanics it's assumed that space is mathematically described as a 3D Euclidean affine space. This enables you to measure distances and angles, areas and volumes etc.

You also need time. By assumption in Newtonian physics it's just a directed 1D axis. At each point you can think to be fastened a copy of the 3D Euclidean affine space, which together gives Newton's absolute space and time.

These are just "educated guesses". You cannot by any mathematical proof be sure that it's correct as a description of the real world. This you can only establish by observations, i.e., being enabled to measure time intervals by clocks and determine the position of points by using e.g., an arbitrary reference point and a cartesian basis describing uniquely each point of space by a position vector or by three Cartesian coordinates wrt. this basis.

Then it's an empirical question, whether your spacetime model is correct. As we know today, it's not entirely correct, because relativistic spacetime models (Minkowski space for SRT and a Lorentz manifold or GR) are valid for all yet known phenomena and not only in the realm of the "non-relativistic limit".

 

[Jiman]

Thank you.The model of space and time is foundation of many other physical problems!