On the difference between mass and weight as it applies to formulae

Question

How exactly does weight work in terms of various formulae include mass? For instance, for the momentum of an object, does momentum increase with greater gravity? Or is it only dependent upon mass ignoring the effects of gravity?

For instance, would applying 1000 Newtons of force on a projectile on Earth, result in it moving at a different velocity than if you used the same force on the same projectile on something with lower gravity, such as the moon, or a planet with higher gravity than Earth.

Another question, following from this, as it is in regards to conservation, if you have two adjacent chambers that somehow have different strengths of gravity, via some sci-fi technology or something, and you shoot a projectile in the higher gravity chamber, would it increase in velocity as it passes into the lower gravity chamber, or would it remain at the same velocity?

In other terms, if an object in motion changes in weight but retains the same mass, would this effect the velocity, or would the fact the mass remains the same mean that it stays the same velocity?

Answer 1

To tell what weight is, its actually the force that mass of a body exerts when an constant force acts on it. For instance when a body of mass 'm' is under free fall, the force acting on the body is gravitational force of earth; hence the weight of the body turns out to be mg, where g is acceleration due to gravity. Here I am considering ideal conditions.

Now coming to your question of momentum. See first of all you cant tell whether momentum increases or decreases due to gravity using Newtonian mechanics, but one thing is for sure that momentum changes. Also presence of friction, gravity and net force results in violation of conservation of momentum. But in problem solving such as collisions we usually ignore gravity and friction hence ,momentum is conserved in TEXTBOOKS.

Your second question's answer is quite simple. I would like to answer it partially and make you think the rest. Now on earth i would give some force to a body, so that is travels like a projectile and fall down. If I increase the force the body would travel further. So if I give it enough force such that, it actually escapes earth's gravity and it eventually starts revolving around the earth (this is the idea of flying rockets). If I increase the force further more then it would completely escape earth's gravity. Now use the same idea for moon or any other planets. Also keep in mind about escape velocity, gravitational potential.

Your third question's answer is again answered in 1st question's answer. Please think more on these. Write down equations of Newtonian Mechanics and also bring in some equations of Relativistic Mechanics.

And your fourth question is not framed well enough. Elaborate it by giving some more constraints, details of the problem.

At last I would like to add something on Conservation Laws: Conservation Laws arise due to some sort of translational symmetry of the universe.

1. Law of Conservation of Momentum arises due to homogeneity of space.
2. Law of Conservation of Energy arises due to homogeneity of time.
3. Law of Conservation of Angular Momentum arises due to isotropic nature of space. When something above is not found, then violation of any of the above can be found.

Hope this helps.

Answer 2

The definition and the interpretation of mass changed through history and between theories. Before diving in these sort of canadrums, you shoud really focus on understanding the meaning of Newton's equation

$$ F = m\times a = m\times \frac{\Delta v}{\Delta t}, $$

that is the indisputable principle of classical mechanics. To answer your question "does momentum increase with greater gravity?", it depends. Since force is the variation of momentum over time, provided that the mass doesn't change, if an object is standing on a surface, nothing is going to happen to it if you suddenly increase gravity. However, for a free falling object the weight is given by $$ m\times g, $$ thus, if you are solid within the Earth system, you will see a greater variation of momentum of the object while it falls.

"would it increase in velocity as it passes into the lower gravity chamber, or would it remain at the same velocity?" There is a serious reason why you cannont build a gravity-based Faraday cage, the reason being that the 'charge', or mass, carrier are not free to move on the boundaries of the chamber, even tho the field equations are practically identical. You may also argue that negative masses aren't a thing, but not only I disagree, I also think that it would be the lesser problem, as we often use negative moments of inertia to compute kinetic energies of complicated objects.

Let's ignore this for a moment, what do you mean by your question? If you mean a locally applied force field then yes, the velocity vector would necessarely change on a free falling object.

" if an object in motion changes in weight but retains the same mass, would this effect the velocity, or would the fact the mass remains the same mean that it stays the same velocity?" Yes, the velocity would change as you observe with orbits. The closer to the Sun, the faster an asteroid must go not to fall on it. Thank God angular momentum is conserved!

You can see this from

$$ \Delta v \propto F(t)\Delta t,\qquad v_f - v_i \propto \sum_{\Delta t} F(t)\Delta t $$

that is, generally, not zero.