Exploring Newton's 2nd Law of Motion

[Antonio Lao]

Is force the cause of motion? If there is motion would there be a force also?

Newton's 1st law of motion says there can be motion even without force. This motion is a constant velocity (in particular such as the constant speed of light in vacuum).

Newton's 2nd law of motion says that the force is proportional to the acceleration with the mass as the constant of proportionality. But if the acceleration is constant then the force is constant for a constant mass system. Constant force is a conservative force. It's time rate of change is always zero.

If the acceleration is taken to be the constant of proportionality and that mass varies then force can be defined proportionally to the mass with the acceleration as the constant of proportionality.

There seem to be two kinds of force implied in Newton's 2nd law of motion depending whether mass or acceleration is taken to be the constant of proportionality.

If both mass (m) and acceleration (a) are not constant then we get

[tex] \frac {dF}{dt} = m \frac {da}{dt} + a \frac {dm}{dt} [/tex]

In the science of rocketry, this equation is taken into consideration.

 

[jdavel]

Is force the cause of motion? If there is motion would there be a force also?

To avoid some confusion, let's change the word "motion" to velocity, the instantaneous time rate of change of a vector from a fixed point to an object. Then the answer to both of your questions is no.

Newton's 1st law of motion says there can be motion even without force.

True (again, with velocity instead of motion)

This motion is a constant velocity (in particular such as the constant speed of light in vacuum).

Well, not really. Newton was talking only about objects with mass. Light doesn't have mass, so it doesn't count.

Newton's 2nd law of motion says that the force is proportional to the acceleration with the mass as the constant of proportionality. But if the acceleration is constant then the force is constant for a constant mass system. Constant force is a conservative force. It's time rate of change is always zero.

Well, that's not what's usually meant by "a conservative force", but you're right, if the mass and acceleration of an object are constant, then the net force acting on the object must be constant.

If the acceleration is taken to be the constant of proportionality and that mass varies then force can be defined proportionally to the mass with the acceleration as the constant of proportionality.

There seem to be two kinds of force implied in Newton's 2nd law of motion depending whether mass or acceleration is taken to be the constant of proportionality.

No, there's only one kind of force. In the equation F=ma, F is the instantaneous force at a particular time t. At that instant, m has some value, and a has some value. Whether one or the other will be different a moment later doesn't affect F and certainly doesn't change in anyway what "kind of force" it is.

If both mass (m) and acceleration (a) are not constant then we get

[tex] \frac {dF}{dt} = m \frac {da}{dt} + a \frac {dm}{dt} [/tex]

In the science of rocketry, this equation is taken into consideration.

That's interesting; I haven't seen it before! If the rocket burns fuel at a constant rate, you could set dF/dt in the equation to zero. Is that how it's used?

 

[Antonio Lao]

I am not sure, I think we need to consult with an expert in rocket technologies.

 

[Antonio Lao]

But I'm sure that the rate of change of force with distance is not zero.

[tex] \frac {dF}{dr} \neq 0 [/tex]

This is an infinitesimal potential energy per unit area or the product of pressure and distance.

 

[Antonio Lao]

Can we also say that the divergence of force is not zero?

[tex] \nabla \cdot F \neq 0 [/tex]

 

[quantumdude]

Constant force is a conservative force.

That's not true.

If a constant force F is applied to a body, its total energy will continually increase (in the form of kinetic energy). What defines a conservative force is whether the work it does on a body is independent of path. Equivalent ways to state this are:

curl(F)=0

and

F=-grad(V),

where V is a potential function.

edit: typo

 

[Antonio Lao]

Can anyone explain to me how the vector potential in EM is related to any type of force? I still don't understand what vector potential is?

 

[quantumdude]

Can anyone explain to me how the vector potential in EM is related to any type of force? I still don't understand what vector potential is?

The vector potential A and the scalar potential F are related to the physical electric and magnetic fields by:

B=curl(A)
E=-grad(F)-(1/c)∂A/∂t

 

[Antonio Lao]

Thank you very much, Tom. Stupid questions: Is this scalar potential independent of path? And vector potental dependent on path?

 

[jdavel]

Antonio Lao asked: "Is this scalar potential independent of path? And vector potental dependent on path?"

They're both independent of path, because they're functions only of postition (x,y,z). You use three equations to calculate A(x,y,z): Ax = f(x,y,z), Ay = g(x,y,z) and Az = h(x,y,z). Nothing in any of the functions f, g or h says how you got to that point, they just say, when you're at that point, here's the value of the function.

The thing that's "path independent" about some (but not) all vector functions, is the line integral of the vector function from one point in space (x1, y1, z1) to another (x2, y2, z2). The condition (as Tom Mattson said) for that line integral of a vector function F to be zero is that curl(F) = 0.

For cases where E and B don't vary with time (static fields) curl(E) = 0, so the line integral of E from one point to another is path independent. For the magnetic field curl(B) = J, so the line integral of B in a region where charge is flowing, is not zero.

 

[Antonio Lao]

jdavel,

Thanks for these valuable elucidations.

 

[jdavel]

jdavel,

Thanks for these valuable elucidations.

You're welcome.

By the way, I learned about electricity and magnetism a LONG time ago and I've actually used them in one way or another pretty continuously since then. But when I see these potentials (even the electric one) I still have to stop and think whether the field is the gradient of the potential, or the other way around around

Maybe this is some kind of mental block (deficiency?) that I have. But I think it's because the way we're introduced to these two potentials is kind of backwards. E = grad(-V) and B = curl(A). It seems to me that an equation defining something should be solved for that thing.

The integral forms for V and A (where they are solved for explicitly) are more intuitively meaningful, at least for me. The equations are almost identical in form, but the difference shows that the fundamental source for a static electric field is charge and that the fundamental source for a static magnetic field is current. So the physics of the potentials seems more obvious in this form.

But grad and curl are powerful tools for doing calculations, because differentiating is often easier than integrating. Thanks to Mr. Gauss and Mr. Stokes, you get to use whichever way is easier for the problem you're working on.

 

[Antonio Lao]

Again, thanks. Elementary particles are known to possesses magnetic moment. magnetic field exists because of motion of charge. The electron is a point particle so how do electric charges move inside the electron?

 

[deda]

Is force the cause of motion? If there is motion would there be a force also?

The force in archimedes's physics is geometrical potential i.e. a storage for the distance yet to be achieved. Because of it as long there is a force there will be motion and without force there cannot be motion. The equation behind this is F * dD > 0. I cannot prove it right but I doubt that anybody can prove it wrong either. It's sort of neutral.

If the acceleration is taken to be the constant of proportionality and that mass varies then force can be defined proportionally to the mass with the acceleration as the constant of proportionality.

Finally somebody considers physics like a real mathematician (like me). I had similar considerations for the definition of velocity dx=Vdt+tdV instead of dx=Vdt

There seem to be two kinds of force implied in Newton's 2nd law of motion depending whether mass or acceleration is taken to be the constant of proportionality.

Newton considers only the mass as constant. Archimedes considers the force - mass ratio as a constant but it is no acceleration at all. In the lever it is : F_1 * M_2 = F_2 * M_1

You kind of insure me that my effort won't be futile.

 

[Antonio Lao]

deda,

Right or wrong, It's good to know that we are in the same wavelength so to speak.