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Online bostonmanTopic starter

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Physics Question - ma = mg
« on: May 13, 2021, 03:03:47 am »
I'm trying to understand exactly why ma = mg.

I took physics, and, the concept of weighing myself on the scale and learning it's really our mass, I'm confused about ma = mg (this began after watching the Big Bang Theory).

If my body mass is 75kg, and I get on a scale, my 'weight' is (75 * 9.8): 735N

If something with a mass of 75kg accelerates at 9.8, then the 'force' is also 735N

Obviously ma = mg in this case. After reading, it's equal because we are on Earth. If I weighed myself on the moon, then ma = mg, however, if everything is relative to gravity on the moon, then ma = mg on the moon. For the most part, ma would equal mg anywhere in the universe providing gravity and acceleration are the same.

Am I missing something in the translation of why ma = mg?
 

Offline Whales

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Re: Physics Question - ma = mg
« Reply #1 on: May 13, 2021, 03:14:55 am »
Algebra:
  m = mass
  a = acceleration
  g = acceleration due to gravity ~= 9.8ms-2 on earth's surface

We start with your eq:

   ma = mg

Divide both sides by m:

   a = g

Yep.  Don't think of g as a variable, but instead an exact value.  Just like pi isn't a variable, it's a value, even though we write it as π instead of 3.14159265.   I can write y = π and I can write a = g; these are both choices I make of subbing in particular values for particular purposes.

g on the moon != g on earth, they're completely different values and hence once subbed in the meaning of the equations gets limited to "only valid on X planet's surface".  Just like how if I choose m=70kg then the above equation becomes "only valid for human beings" instead of "valid for everything with a mass in a gravitational field". 

Does this help, or am I going down completely the wrong path?
« Last Edit: May 13, 2021, 03:19:25 am by Whales »
 
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Offline andy3055

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Re: Physics Question - ma = mg
« Reply #2 on: May 13, 2021, 03:22:01 am »
Here is a good page to read (ThoughtCo.com): https://www.thoughtco.com/mass-and-weight-differences-606116
 
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Online bostonmanTopic starter

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Re: Physics Question - ma = mg
« Reply #3 on: May 13, 2021, 03:38:12 am »
Quote
g on the moon != g on earth, they're completely different values and hence once subbed in the meaning of the equations gets limited to "only valid on X planet's surface".

That was my point. My 'mass' (using 75kg) is the same throughout the universe and can't change (unless I eat lots of food while venturing around space). a = g doesn't only exist on Earth, but can exist anywhere in the universe too.

So saying because we are on Earth, and acceleration due to gravity is 9.8, then ma = mg, isn't really correct.

 

Offline CatalinaWOW

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Re: Physics Question - ma = mg
« Reply #4 on: May 13, 2021, 03:44:27 am »
Quote
g on the moon != g on earth, they're completely different values and hence once subbed in the meaning of the equations gets limited to "only valid on X planet's surface".

That was my point. My 'mass' (using 75kg) is the same throughout the universe and can't change (unless I eat lots of food while venturing around space). a = g doesn't only exist on Earth, but can exist anywhere in the universe too.

So saying because we are on Earth, and acceleration due to gravity is 9.8, then ma = mg, isn't really correct.

You are right.  The equality is not general, it only holds for the special case where a=g.  That special case does apply almost universally on this planet and as far as we know no one lives anywhere else (space station visits not counting as living), so you can see where someone got a little ahead of themselves with this one.
 
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Offline Whales

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Re: Physics Question - ma = mg
« Reply #5 on: May 13, 2021, 03:55:57 am »
So saying because we are on Earth, and acceleration due to gravity is 9.8, then ma = mg, isn't really correct.

It is correct, but only in certain situations.  Basically no equation is truly universal, all have limitations.

   f = ma  <-- only directly usable as an approximation between a very large mass (earth) and a very small mass (eg a person) OR between two very distant bodies where their radii are much less than their seperatory distance.  Otherwise the results are quite wrong (you have to break each mass down into smaller pieces & integrate the results instead).

   f = m * 9.8ms-2   <--- all of the above restrictions, but now also only valid on the earth's surface (and technically in some other places in the universe, but this explanation is good enough)

   f = 75kg * a  <--- only valid for things of that weight, eg approx a human

  f = (width * length * height * density) * a  <-- only valid for rectangular prisms of constant density

Online Anthocyanina

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Re: Physics Question - ma = mg
« Reply #6 on: May 13, 2021, 04:01:29 am »
It's important to understand the distinction between mass and weight. Your weight on the moon is different because gravity on the moon is a different value than gravity on earth. weight is the force exerted by a mass because of gravity. gravity on earth is called "normal gravity" and the mass is calculated according to earth's gravity, so weight on earth is the same as mass on earth. mass on the moon is the same as mass on earth because the matter that gives it that mass would be the same matter on earth (and now under the influence of the "normal gravity") and its mass is calculated with our planet's gravity
 

Offline IanB

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Re: Physics Question - ma = mg
« Reply #7 on: May 13, 2021, 04:10:40 am »
So saying because we are on Earth, and acceleration due to gravity is 9.8, then ma = mg, isn't really correct.

It is correct, but only in certain situations.  Basically no equation is truly universal, all have limitations.

   f = ma  <-- only directly usable as an approximation between a very large mass (earth) and a very small mass (eg a person) OR between two very distant bodies where their radii are much less than their seperatory distance.  Otherwise the results are quite wrong (you have to break each mass down into smaller pieces & integrate the results instead).
Excuse me? F = ma is one of the fundamental laws of Newtonian mechanics. It is always exactly true, without exception, in a Newtonian framework. What makes you think it is an approximation?

Quote
   f = m * 9.8ms-2   <--- all of the above restrictions, but now also only valid on the earth's surface (and technically in some other places in the universe, but this explanation is good enough)

   f = 75kg * a  <--- only valid for things of that weight, eg approx a human

Again, what the heck? F = mg is universally true for any object of any mass in a gravitational field. The thing to remember is that g varies with the magnitude of the gravitational field, so it is almost never exactly 9.81 m/s2, although it is approximately that on the Earth's surface.

Quote
  f = (width * length * height * density) * a  <-- only valid for rectangular prisms of constant density
This is just a total non-sequitur. It is technically true in some sense, but has no useful meaning.
 

Offline IanB

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Re: Physics Question - ma = mg
« Reply #8 on: May 13, 2021, 04:17:35 am »
I'm trying to understand exactly why ma = mg.

Well, it's not, in general, true. So maybe that's why you are confused?

What is true is that mg = mg (obviously). The symbol "g" is just used for the special value of acceleration that applies to a free falling body in a given gravitational field (usually Earth's). In this case the acceleration experienced by that body is given by the symbol "g". In general, "a" can have any value at all, but in the specific case of a free falling body in Earth's gravitational field, "a" = "g". So F = mg is just a special and unique case of the more general F = ma. That's all there is to it.
 
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Offline AntiProtonBoy

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Re: Physics Question - ma = mg
« Reply #9 on: May 13, 2021, 04:18:47 am »
Obviously ma = mg in this case. After reading, it's equal because we are on Earth. If I weighed myself on the moon, then ma = mg, however, if everything is relative to gravity on the moon, then ma = mg on the moon. For the most part, ma would equal mg anywhere in the universe providing gravity and acceleration are the same.

Am I missing something in the translation of why ma = mg?

Weight and inertial mass are different things. Weight and force are also different things, but closely related.

The relationship F = ma simply tells you what is the resulting force F exerted by mass m when accelerated at a given rate a.

In the context of Earth, gravity gearth is a special case of acceleration a = 9.81. In this case, your weight is related to the force exerted by your body's mass when accelerated by gravity. We standardise this weight such that 1 kg = 9.81 newtons.
In other words, weightearth = m * a / 9.81. Since a = gearth on earth, weight and mass is equivalent in magnitude, that is weightearth  = m.

On the moon, your mass would still remain the same, but the weight would be less due to different acceleration by gravity, a = gmoon = 1.63.
In this case, weightmoon = m * 1.63 / 9.81 = m * 0.17.

TL;DR:

Generic Newton's force formula:

F = m * a

Weight force on any planet (substitute a with gravitational acceleration):

Fplanet = m * gplanet

Weight on any planet in kg:

weightplanet = m * gplanet / 9.81

« Last Edit: May 13, 2021, 04:21:23 am by AntiProtonBoy »
 
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Offline T3sl4co1l

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Re: Physics Question - ma = mg
« Reply #10 on: May 13, 2021, 04:21:53 am »
IIRC, it's surprisingly not well understood why inertial mass equals gravitational mass, at least to high observational precision (which isn't all that high in the grand scheme of physicsthings, because gravity is difficult to measure).

The reason given by General Relativity, is that space itself is moving, accelerating towards gravitational wells.  There was an excellent animation of this in a recent YT video, which of course I didn't bookmark, so if someone can remember it please put it here -- in any case, the fact that spacetime itself is curving in towards a mass, is equivalent and indistinguishable to inertial mass being accelerated.

What's weird about the picture is not that space is accelerating, but why we aren't when we stand on the Earth!  Well, the answer is relative, of course -- since space is moving past us here on the surface at ~9.8 m/s^2, something must be accelerating us up through all of it.  The force for that eternal acceleration is -- drumroll -- the force of your weight pushing you along, and so on down to the core of the Earth where all the force, from the entire mass of the planet, is bearing against itself; fortunately, planetary matter is not very compressible at these pressures (but it isn't resistant to shear, and so the Earth gets pulled into a sphere, more or less -- it reaches hydrostatic equilibrium).

So, obviously that weight exactly equals your inertial weight, because otherwise you'd be dragged along in space at 9.8 m/s^2 relative to the Earth's surface.


There are some weird ideas, like Mach effects, which I don't think have been disproven?  Or perhaps they're equivalent after all, but we don't quite yet understand how.  The deeper mystery is how to integrate General Relativity into the Standard Model, so that we have a complete understanding from the smallest quantum level to the scale of the entire universe.  Perhaps then we will have a more complete explanation.  That, however, will take some time it seems. :)


Also:
Purely from Newtonian mechanics, there is no answer, of course.  Newton's laws can be derived from Relativity given suitable approximations (c --> infty), just as Relativity is an approximation of some as-yet-unknown better model, and so on (it's turtles all the way down, at least until we have a more convincing reason to believe otherwise).

Tim
« Last Edit: May 13, 2021, 04:26:05 am by T3sl4co1l »
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Offline thermistor-guy

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Re: Physics Question - ma = mg
« Reply #11 on: May 13, 2021, 04:38:07 am »
I'm trying to understand exactly why ma = mg.

I took physics, and, the concept of weighing myself on the scale and learning it's really our mass, I'm confused about ma = mg (this began after watching the Big Bang Theory).
...

I read your "ma=mg" as indicating inertial mass and gravitational mass are the same:

https://www.einstein-online.info/en/spotlight/inertial-and-gravitational-mass/

"Gravity"describes how bodies move through space-time, which is curved by large masses. If you treat gravity as a force, then it turns out that the acceleration of small bodies are the same regardless of mass. This is peculiar when you think about it.

Go to the gym. Lift a 10 kg mass, then a 20 kg mass with the same acceleration. The 20 kg weight takes more effort (force) on your part.

Drop the 10 kg and 20 kg weight from shoulder height to the gym floor. The two weights move through space-time, with the same acceleration. If you regard gravity as a force, then "gravity" must be exerting more force on the 20 kg weight - twice as much - as on the 10 kg, the same way you did during your lifts. This is odd.

Now imagine jumping out a window with the 10 kg weight and releasing it mid-air. You and the weight are travelling with the same acceleration through space-time. But you don't feel any force on you, you don't feel any pressure on your body, the way you do when you lift a weight, or go up in a fast elevator. Is gravity really a force, then? If it is, it's a strange kind of force where you don't feel any :-)

Interestingly, near the Earth, space-time is curved mostly by the Earth, but there is also a tiny effect due to the sun and moon. An extremely sensitive pendulum clock will detect it:
https://en.wikipedia.org/wiki/Shortt%E2%80%93Synchronome_clock
http://leapsecond.com/pend/pdf/1986-Mar-AH-Boucheron-Shortt.pdf

Physicists aside, we treat gravity as a force because it matches our everyday language, and makes sense of our everyday experience. But it is a misleading line of thought, according to General Relativity.
https://www.universetoday.com/108740/how-we-know-gravity-is-not-just-a-force/
 

Offline T3sl4co1l

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Re: Physics Question - ma = mg
« Reply #12 on: May 13, 2021, 04:52:09 am »
Physicists aside, we treat gravity as a force because it matches our everyday language, and makes sense of our everyday experience. But it is a misleading line of thought, according to General Relativity.
https://www.universetoday.com/108740/how-we-know-gravity-is-not-just-a-force/

And (not to be redundant as I haven't read the link to see if it happens to make this connection already..!) we have a similar case with "centrifugal force": the apparent force to the outside, is actually the rotating object pulling inward.  The difference between centrifugal and centripetal is merely a coordinate transformation.

...And no Mr. Bond, I expect you to die.

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Offline Brumby

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Re: Physics Question - ma = mg
« Reply #13 on: May 13, 2021, 07:21:03 am »
I am consistently amazed at the tangents and minutiae that appear in response to such a simple question.  Some have made good effort at trying to clarify while others have been anything but.

This, to me, is the best offering so far:
Generic Newton's force formula:

F = m * a

Weight force on any planet (substitute a with gravitational acceleration):

Fplanet = m * gplanet
This is where a general formula benefits from subscripts to distinguish different scenarios.
For example:
     gearth
     gmoon
     gmars
     gpluto
     goumuamua
All these values for g will be different - but they plug into equations the same way - giving results that are correct for their respective situations.

But then it is spoiled by this:
Quote
Weight on any planet in kg:

weightplanet = m * gplanet / 9.81
For starters, kg is a unit of mass - not a weight.  Secondly, what is that divide by 9.81 all about?
« Last Edit: May 13, 2021, 07:28:07 am by Brumby »
 

Offline thermistor-guy

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Re: Physics Question - ma = mg
« Reply #14 on: May 13, 2021, 12:47:33 pm »
I am consistently amazed at the tangents and minutiae that appear in response to such a simple question....

I don't regard it as a simple question. To me, the equivalence principle (inertial and gravitational masses are equivalent) is profound and mysterious.

Yeah we can do simple calculations, but lurking behind the "why" of those calculations lies something deep about the nature of reality. Since the OP has studied physics, I assumed the question was about more than simple calculation, and more about the "why". Looks like I've misunderstood.
 

Offline TimFox

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Re: Physics Question - ma = mg
« Reply #15 on: May 13, 2021, 01:33:49 pm »
One of the great 19th-century physics experiments was carried out by the Hungarian Baron Eötvös from 1885 to 1909.  See https://en.wikipedia.org/wiki/E%C3%B6tv%C3%B6s_experiment
His group showed that gravitational and inertial mass of different materials was equivalent to a sensitivity of about 10^-8.
Note that g, the gravitational acceleration due to the earth, does change with altitude, and slightly due to variations in the density underfoot, but for most purposes (except for inertial-guidance navigation) it is a constant vector pointing down.
Kater's pendulum is a classic way to measure g  https://en.wikipedia.org/wiki/Kater%27s_pendulum
In old laboratory textbooks, the period of the pendulum was measured by having it oscillate in front of a grandfather clock and comparing the frequency to that of the clock pendulum (illustration in above Wikipedia reference).
 

Offline mathsquid

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Re: Physics Question - ma = mg
« Reply #16 on: May 13, 2021, 04:12:26 pm »
Rather than going about defining different subscripted g's, I'd use the law of universal gravitation and G = 6.67408 x 10-11 m3 kg-1 s-2

$$ F= G \frac{m_1m_2}{r^2}$$

Plugging in mass and mean radius of the earth gives you

$$ F = 6.67408 \cdot 10^{-11}  \, \frac{5.971 \cdot 10^{24} m}{(6.371\cdot 10^6)^2} \approx 9.8m$$

meters/s2
« Last Edit: May 13, 2021, 04:16:18 pm by mathsquid »
 

Offline radiolistener

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Re: Physics Question - ma = mg
« Reply #17 on: May 13, 2021, 06:50:28 pm »
Obviously ma = mg in this case.

This is not obvious. This is known as equivalence principle, which says that the gravity mass equals to the inertial mass.

From experiments we know that the gravity mass is very close to the inertial mass, the difference is less than 10^-14 or something like that. But we don't know if they are equals exactly or there is a very small difference which we cannot catch due to insufficient measurement resolution. And there is no explanation why they are very close.

Nobody knows that. So, this is very not obvious.  :)

Probably you will be interested to read the Chapter 28 "Electromagnetic mass" in The Feynman Lectures on Physics Volume 2.
It is about the mass of electron. A little quote:
Quote
We only wish to emphasize here the following points: (1) the electromagnetic theory predicts the existence of an electromagnetic mass, but it also falls on its face in doing so, because it does not produce a consistent theory—and the same is true with the quantum modifications; (2) there is experimental evidence for the existence of electromagnetic mass; and (3) all these masses are roughly the same as the mass of an electron. So we come back again to the original idea of Lorentz—maybe all the mass of an electron is purely electromagnetic, maybe the whole 0.511 MeV is due to electrodynamics. Is it or isn’t it? We haven’t got a theory, so we cannot say.

We must mention one more piece of information, which is the most annoying. There is another particle in the world called a muon—or μ-meson—which, so far as we can tell, differs in no way whatsoever from an electron except for its mass. It acts in every way like an electron: it interacts with neutrinos and with the electromagnetic field, and it has no nuclear forces. It does nothing different from what an electron does—at least, nothing which cannot be understood as merely a consequence of its higher mass (206.77 times the electron mass). Therefore, whenever someone finally gets the explanation of the mass of an electron, he will then have the puzzle of where a muon gets its mass. Why? Because whatever the electron does, the muon does the same—so the mass ought to come out the same. There are those who believe faithfully in the idea that the muon and the electron are the same particle and that, in the final theory of the mass, the formula for the mass will be a quadratic equation with two roots—one for each particle. There are also those who propose it will be a transcendental equation with an infinite number of roots, and who are engaged in guessing what the masses of the other particles in the series must be, and why these particles haven’t been discovered yet.
« Last Edit: May 13, 2021, 07:21:05 pm by radiolistener »
 

Offline TimFox

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Re: Physics Question - ma = mg
« Reply #18 on: May 13, 2021, 07:54:54 pm »
Obviously, the gravitational and inertial mass can be defined as equal for one specific material, such as pure gold.  Baron Eötvös showed that this was true (within very small experimental error) for a wide range of materials.  My favorite statement (attributed to) Albert Einstein is "Raffiniert ist der Herrgott, aber boshaft ist er nicht,”.  Surely only a demon would violate the equivalence principle by the teeny margin of 10^-14.
 

Offline AntiProtonBoy

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Re: Physics Question - ma = mg
« Reply #19 on: May 14, 2021, 01:32:52 am »
But then it is spoiled by this:
Quote
Weight on any planet in kg:

weightplanet = m * gplanet / 9.81
For starters, kg is a unit of mass - not a weight.  Secondly, what is that divide by 9.81 all about?

kg and unit of mass equivalence is only true on Earth. In other words 1 kg of weight equals to 1 unit of mass exerting exactly 9.81 newtons of force on earth; or alternatively we can also say that 1 kg of weight equals to 1 unit of mass accelerated by 9.81 m/s2. If you want to measure weight on another planet, then you will need to scale by the ratio gplanet / 9.81.

Example: if you weight 100 kg on earth,  then you will weigh 16.53 kg on the moon. Here is why:

weightearth = m = 100 kg
gmoon = 1.62

weightmoon = m * gmoon / 9.81
weightmoon = 100 * 1.62 / 9.81
weightmoon = 16.53 kg

Wiki has a pretty good explanation on this:

In scientific contexts, mass is the amount of "matter" in an object (though "matter" may be difficult to define), whereas weight is the force exerted on an object by gravity. In other words, an object with a mass of 1.0 kilogram weighs approximately 9.81 newtons on the surface of the Earth, which is its mass multiplied by the gravitational field strength. The object's weight is less on Mars, where gravity is weaker, and more on Saturn, and very small in space when far from any significant source of gravity, but it always has the same mass.

https://en.wikipedia.org/wiki/Mass_versus_weight


Also, how do I insert LaTeX formulas in this forum?
« Last Edit: May 14, 2021, 01:35:32 am by AntiProtonBoy »
 
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Offline JohnnyMalaria

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Re: Physics Question - ma = mg
« Reply #20 on: May 14, 2021, 01:58:59 am »
So saying because we are on Earth, and acceleration due to gravity is 9.8, then ma = mg, isn't really correct.

It is correct, but only in certain situations.  Basically no equation is truly universal, all have limitations.

   f = ma  <-- only directly usable as an approximation between a very large mass (earth) and a very small mass (eg a person) OR between two very distant bodies where their radii are much less than their seperatory distance.  Otherwise the results are quite wrong (you have to break each mass down into smaller pieces & integrate the results instead).
Excuse me? F = ma is one of the fundamental laws of Newtonian mechanics. It is always exactly true, without exception, in a Newtonian framework. What makes you think it is an approximation?

Your own assertion answers your question :)

It's an approximation because it's only "exactly true" "in a Newtonian framework". Ergo, it's an approximation because it fails outside of a Newtonian framework.
 

Offline mathsquid

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Re: Physics Question - ma = mg
« Reply #21 on: May 14, 2021, 02:02:43 am »
Also, how do I insert LaTeX formulas in this forum?

To do displayed math, just put it between double dollar signs. I don't know if there's a way to do it inline.
 
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Online bostonmanTopic starter

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Re: Physics Question - ma = mg
« Reply #22 on: May 14, 2021, 03:13:53 am »
I'm somewhat glad my question wasn't so easily answered because I thought it was too simple. As someone pointed out above, some of this has deviated and dove deep into physics.

Although the basic plug and chug formulas make things easy, sometimes they don't make sense after trying to analyze them.

One thing we can all agree on is mass remains constant, it can't change throughout the universe. If I want to know my weight on Earth, using 100kg for Mass to make the math easier, then (W= mg) it's 9.8 * 100kg = 980N. If a 100kg object accelerates at 9.8m/s^2 for 1s, then the Force (F = ma) would also be 980N.

Therefore ma = mg.

Now take the same situation above and place me on the moon. My weight is 162N (moon gravity = 1.62m/s^2) and the 100kg object accelerating at 1.62m/s^2 for 1s is also 162N.

Therefore ma = mg.

My point is: from my understanding, no special case exists where ma = mg because we are on Earth. From my initial assumption, it seemed a special case existed where ma = mg because we are on Earth and/or a connection existed with 9.8 to make ma = mg.

Basically, they can equal each other anywhere in the universe providing the accelerating object accelerates at the same rate as the gravity as to where you are.

I don't want to deviate, but I'm a bit uncertain how F=ma and W=mg were discovered. We can calculate Mass because we know gravity, but centuries ago, nobody knew the acceleration of gravity, so they couldn't calculate the Mass being dropped. It's kind of the chicken or the egg I guess.

Also, if I place a 100kg block on a table, is the block a Weight with the table acting as a Force in the upward direction, or is the block also a Force? During this discussion, I began realizing, I'm confused as to when to use a Weight and when to use a Force (unless we are talking about say a car driving into an object, then that's a Force).

For reference, at 5:20 is where the original question (from The Big Bang Theory) came from:

 
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Offline AntiProtonBoy

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Re: Physics Question - ma = mg
« Reply #23 on: May 14, 2021, 03:19:16 am »
To do displayed math, just put it between double dollar signs. I don't know if there's a way to do it inline.

Inline test: \$ foo = x \$

$$ foo = x^6 $$

Edit: looks like inline uses the slash dollar sign syntax.
Also LaTeX doesn't work when previewing the post. Dave, is there way to fix this?
« Last Edit: May 14, 2021, 03:21:06 am by AntiProtonBoy »
 
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Offline Alex Eisenhut

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Re: Physics Question - ma = mg
« Reply #24 on: May 14, 2021, 03:40:49 am »

One thing we can all agree on is mass remains constant, it can't change throughout the universe.

Not sure that's correct. There's rest mass and relativistic mass.

https://en.wikipedia.org/wiki/Mass_in_special_relativity
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