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Physics Question - ma = mg

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Nominal Animal:

--- Quote from: bostonman on June 17, 2021, 02:24:12 pm ---What I make of ma=mg is that an object sitting on the ground is experiencing a force F=ma where 'a' is Earth's gravity, and its weight is W=mg where 'g' is gravity on Earth; therefore ma = mg.

--- End quote ---
This is the correct intuitive understanding, I believe.


Mass is the property of matter that does not depend on gravity.
We call the force due to gravity 'weight', and that does depend on gravity; it is that force that weighing scales etc. measure.

On other planets and moons, 'g' differs, so objects have different weights there, despite having the same mass.

On Earth, it takes 0.45 seconds to drop 1 meter from standstill, because g = 9.80 m/s².
On the Moon, it takes 1.11 seconds to drop 1 meter from standstill, because there g = 1.62 m/s².

A one liter bottle of water weighs 1.00 kg on Earth, but only 0.165kg on the moon.
Its mass m=1.00 kg everywhere, because to accelerate it at 1 m per second per second, 1 m/s², you need a force of 1 N, if we ignore possible drag and friction, on Earth as well as the Moon.
It is only the downwards force, 'weight', that varies due to different gravities.

Physicists use an air cushion, like an air hockey rail, to create a very nearly frictionless surface to slide tiny carriages on.
This is a very illustrative device, giving the correct intuition in this matter in a practical way.
No matter in what kind of gravity we are in (as long as the air cushion rail still works), the same mass needs the same force to accelerate at the same rate.
It is only the 'weight' and not 'mass' that is affected by gravity.

Nominal Animal:
In case someone is wondering whether 'mass' itself could actually be split into two different quantities based on what I wrote above, wonder no longer: Even Einstein contemplated that.

The two "different" aspects of mass are called gravitational mass ("slow mass") and inertial mass ("fast mass").  One is the quantity related to gravity (the 'm' when dealing with weight); and the other is the one related to movement and inertia (the 'm' when dealing with actual acceleration).

Experiments on Earth and in Earth orbit (microgravity) have shown that the two must be equal to at least 12 significant decimal digits: that even if they were different, their relative difference must be less than 1:1,000,000,000,000.  So, for all intents and purposes in the human scale, there is just one 'mass' for each object.

A core starting concept in Einstein's theory of general relativity is the equivalence principle: that gravitational mass and inertial mass are equal and indistinguishable.
It actually goes even further, as the strong equivalence principle states that locally, acceleration and gravity are indistinguishable.  (That in a small closed box, given any scientific instruments and experiments you want, it is impossible to distinguish whether the box is standing still on a planet, or in constant acceleration.)
All experiments we have managed to devise thus far indicates these are exactly, precisely true.

bostonman:
Do we ever really care about Weight in equations unless we want to know based on what an object is on planets?

Another words, if a box is resting on a table, it's just all based on Forces, correct? It would be the force in the upward direction the table exerts, and the force in the downward direction the box exerts. Tilt the table 45 degrees, and now there is a X and Y axis force (the force the box has in the X direction and friction in the X direction opposite the box), and a Y, with a Fnet.

Do we care what the Weight is in at all?

Nominal Animal:

--- Quote from: bostonman on June 18, 2021, 02:16:04 pm ---Do we ever really care about Weight in equations unless we want to know based on what an object is on planets?

--- End quote ---
We derive friction from weight, but that's about it.


--- Quote from: bostonman on June 18, 2021, 02:16:04 pm ---Another words, if a box is resting on a table, it's just all based on Forces, correct?
--- End quote ---
Correct.


--- Quote from: bostonman on June 18, 2021, 02:16:04 pm ---Tilt the table 45 degrees, and now there is a X and Y axis force (the force the box has in the X direction and friction in the X direction opposite the box), and a Y, with a Fnet.
--- End quote ---
An illustration should be in order here.  The three forces shown are vector quantities, with direction and magnitude.


Red Fw is the force due to gravity the box exerts on the tilted surface.
Blue Ft is the static force the surface exerts on the box, because it resists deformation.
Green Ff is the force due to friction.

If the velocity of the box does not change, then Fw + Ft + Ff = 0.
If the velocity of the box does change, then Fw + Ft + Ff = m a, where m is the mass of the box, and a is the acceleration vector.  Its direction is opposite to friction, Ff.

(This assumes perfectly smooth but not frictionless surface, and assumes the friction and tilt angle are such that the box does not start tumbling.  If we'd add angular momentum, we'd see that the friction causes torque, which causes the leading edge of the box to exert larger force than the trailing edge of the box, and even a minor imperfection in the surface can catch the leading edge, increasing the forces such that the box starts rolling.  Also, if the center of mass is not supported by the surface the box is on, it will tumble.)

(If we add momentum and angular momentum, we have a pretty accurate simple physics engine at hand.)

So, if we are trying to model the movement of an object, we care about weight, because it relates to friction and the stress on materials holding it in place (like the table surface above).  To calculate acceleration, we use mass.

Static friction is basically always larger than dynamic friction.  In other words, if the object is not moving with respect to the surface it is on, the friction is much greater than when the object is moving with respect to the surface.  This is why objects on tilted surfaces often only need a small nudge, but then keep moving, often accelerating.  The initial nudge overcomes the static friction, and the dynamic friction is not large enough to stop the movement.
Aside from this, it is important to realize that forces cause acceleration; without forces, velocity stays constant.

If we push the abovementioned box to some speed on an inclined surface, and the friction is larger than the other forces combined, then the box slows down: it decelerates (decelerates == acceleration vector opposite to its velocity vector).  When the box stops, the friction is exactly the other two forces combined.  Because static friction is greater than dynamic friction (consider it a curve that drops quickly to a near constant, as velocity increases), there is often a bit of a jerk at the end.
If the box "bounces" backwards a bit, that is due to temporary deformations relaxing; essentially spring-back.

RJSV:
Ok, uh BRUMBY: (thank you)
   Is it alright, if my HEAD HURTS now ??  Jeez, but you make the best summary...I agree but interesting to read.
Question about 'g' is that it's weird to call it acceleration directly. Perhaps it just has exact same units, being in meters per second, per second. So an object, on a scale, is not moving, per SE, but rather it just calculates out in a very similar math process.
   How did Newton prove (or did he?)  a derivation ?
I did college level 'classic' physics. Did Newton use integration, using another formula first ?
  I find I can't clearly address that...

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