# Why do gravitational mass and inertial mass appear to be indistinguishable? [duplicate]

I have learnt that heavier the object is (the more gravitational mass it has), the more resistance to the change of motion it is (the more inertial mass it has).

I can accept this fact but I can't find out the reason behind it. What dynamic, what phenomena could cause this? Does it have something to do with the atomic structure of the object?

## marked as duplicate by stafusa, knzhou, Qmechanic♦Jan 26 '18 at 10:04

The answer is that more mass is defined to provide more inertia.

Newton noticed that, for any given object, $F\propto a$, that is to say the force on an object and the acceleration of that object are proportional. Whenever we find a proportionality like this, we assign a multiplier to turn that $\propto$ into an $=$. Thus we have $F=ma$. Mass is defined to be the constant of proportionality that converts accelerations into forces.

Once you define mass as such, you can rearrange the equation to $a=\frac{F}{m}$, and that shows how if you push on a more massive object with a specified force, the acceleration is smaller than if you pushed on a less massive object. This is true simply because we defined the idea of "massive" around this equation.

• Comments are not for extended discussion; this conversation has been moved to chat. – David Z Jan 29 '18 at 5:12

Picture it like this. Imagine you have 2 crates. In each crate there is pure iron. But one box has 2 times the number of iron atoms, so it is has twice the mass. The weight of box should be negligable.

Now imagine that they are both moving at the same speed. When you apply same force on both boxes, the box with twice the number of iron atoms is being slowed less (has bigger mass) because you have to stop each atom of iron that has it's own same kinetic energy compared to individual iron atoms in the other box. This means that you will not be able to slow them all as much, as you could slow just a few iron atoms in the other box. This is the reason behind inertia.

Each iron atom at certain speed has the same kinetic energy as every other iron atom (If you don't take into account termal motion). This means that in bigger box, you have to add up all the iron atoms and multiply it with their movement energy. Because there are twice as many in other box (twice the mass) it will be harder to stop them all.

• I think this is the answer the OP is looking for, even if it could be worded more elegantly (for example, I'm not sure whether it's necessary to go into atoms, although it is perfectly correct). The essence is that if you have n identical objects, all the properties multiply by n. The abstraction that $n \in \mathbb{Q}$ seems not difficult. – Peter A. Schneider Jan 26 '18 at 8:37
• There is a weak point: as mentioned by P. A. Schneider you are using "number" argument for identical objects. Is is however unclear why the gravitational mass and inertial mass should be equal if there are two object of different in nature (neutrino and quark). Why is the proportionality constant between gravitational and inertial mass in both cases the same (by convention equal to 1)? – F. Jatpil Jan 26 '18 at 9:52

There is no apriori reason in classical mechanic for the gravitational mass to be the same as inertial mass. Is is just an observation: the two constants appear in completely different context (gravitational force formula vs. acceleration formula) and classical mechanic does not provide a reason for them being the same (or proportional with a universal constant).

The answer is obviously general relativity. There they are the same because "gravitational" mass disappears in general relativity. All what remains in general relativity is inertial mass. Bodies move in straight lines (they do not stop, because they have inertial mass) in curved time space and there is no "gravitational force" per se. Of course Einstein constructed the theory in this way intentionally in order to exactly do this. Well: the complication to pay is a curved space-time... So one could actually ask: "why is the specetime-curving constant equal to the inertial mass?" (It seems to me that Einstein just re-labeld the problem but did not solve it.)

So the answer: our best model says there is actually no "gravitational mass".

I would say just take Newton’s second law. F(net) = m a

Let’s say there are two masses m1>m2

To bring equal acceleration in the two blocks,Force applied should be:-

F1= m1 a

F2 = m2 a

By simple mathematics,you know F1> F2

This difference is,rather called, Inertia

I’m a newbie in Physics so can’t provide atomic level explaination,I have just shown this concept in a satisfactory but not adequate or widely accepted or highly advanced terms. And yeah I have just said that the difference is called inertia,that’s just an assumption by me and not the exact definition of the term