I get **ΣF = m*a** and gravity pull is **m*g**
But I was once told that there is no reason that the mass in both cases are the same. However, they happen to be according to our most precise measurement. Why wouldn’t they be the same? Can anything have different mass in regard of the context?
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Gravitational Mass is just a specific application of Inertial Mass. Inertial Mass is more general and can be mass under any force – gravity is a special case of it.
Look at it that way:
Inertial mass is how much an object resists to a force (any force) trying to change its velocity.
Charge is how much an object affects other charged objects via electromagnetic forces.
Gravitational mass is how much an object affects other massive objects via gravitational forces.
Charge is not related to inertial mass. Why would gravitational mass be related to inertial mass (beyond the fact that they use the same unit)?
Of course, experimentally, we ended up noticing that both masses have the same value for every object and deeper understanding of gravity (GR) actually removed that duality altogether.
Your question is vague. Are the two formulas depicting the same object in the same situation? Then the answer is an absolute yes, except if it’s a continuum of great size.
In general, of course, those two can be describing completely different objects and situations.
In understanding that question, it helps to start with the full version of Newton’s second law which is:
ΣF=dp/dt where p is momentum (m•v)
by chain rule that is:
ΣF=m•dv/dt + v•dm/dt
In most cases, the mass of the object you are thinking of doesn’t change (dm/dt=0). So this reduces to:
ΣF=m•dv/dt=m•a
Some problems, however, require considering the full version. Think of a rocket leaving the surface of the planet. Maybe 2/3 of its mass is fuel. So as it is being propelled upwards, it is quickly losing mass, and any prediction of its motion that doesn’t take that into consideration will be incomplete.
Also, and this might be more to the point of what you were asking, there is the issue of center of mass vs. center of gravity. Usually these are the same; each particle of your body feels about the force due to gravity so the weighted center of mass and weighted center of gravitational force will be in the same spot. But if, for some reason, you were thinking of moving feet first toward a black hole, then the gravitational force changes so quickly that the force felt by the particles in your feet is much stronger than the force felt by the particles in your head. So in that case, your center of gravity would be much closer to your feet than your center of mass which remains in the same place as it would be if you were standing on the surface of the Earth or anywhere else.
As a side note; having done a similar problem in grad school, the gradient of the gravitational force as you approach a black hole is so steep, that it would pull the individual particles that you are made of apart long before you reached the event horizon. Because two particles, even though they are very close together, experience such disparate forces that one will be pulled off of the other one, overcoming the bonds that are keeping them together.
We have never found any material which is effected by gravity differently from force. Some objects change mass over time, like rockets, but that doesn’t count.
There might be things like that, think dark matter, but there is no real definition of that which has been well supported by evidence.
If you have mass then your gravitational mass and inertial mass is the same. Of course, your gravitational mass can only be measured here on Earth, so that’s not super general.
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