Why does the mass of both objects not factor? Did I misunderstand the reason for differing gravity on other planets?
In: Physics
The acceleration due to gravity depends on the mass of the object that is being accelerated both as a resistance to said acceleration (inertia) and as a cause of the force. That means that there are two “opposing” effects, which cancel out perfectly. With some algebra, this can be shown very easily:
F = ma
GMm/r^2 = ma
GM/r^2 = a
It does! Sort of. The mass of the object you drop decides how fast Earth falls *towards it*. That speed is just going to be really *really* slow no matter what you drop.
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The force of gravity does depend on the mass of the two things, like you said. Earth’s gravity does pull harder on a 10 lb steel ball than a 1 lb wood ball.
**But it also takes more force to accelerate a heavier object!**
These two things cancel out. Gravity pulls the steel ball harder, but it takes more pull to accelerate the heavy ball. The way the math works out, these two factors always exactly cancel out so that the resulting falling speed is always the same, and doesn’t depend on the weight of the dropped object (ignoring air resistance).
See my answers from earlier today:
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byu/Panchito135 from discussion
inexplainlikeimfive
It does, but it’s only noticeable if both objects are massive.
Force = Gravitational constant * mass1 * mass2 / radius^2.
That force works both ways. The Earth pulls a hammer at the same acceleration as a feather, but the earth is accelerated faster towards the hammer than to the feather. It’s just that a force equivalent to the weight of a hammer is still negligible when compared to the mass of the Earth, even if it’s hundreds of times higher than the feather.
on planets both distance from the center of gravity and the mass of the two objects are effectively constant when working on human scales.
it’s a bit simpler if we look at these two formulas.
the gravitation formula is as follows: G * (m1 * m2) / (r^2)
G is a constant so we can ignore it. m1, m2 are the masses of the two objects that attract each other (earth and a ball for instance). r is just the distance between the objects.
if you look at this, we can see that it depends on the mass.
we can confirm this by taking into consideration the gravitational force of a black hole
and the formula for speed, taking into consideration the formula of speed, based on acceleration and original speed: s1 = s0 + a * t
we can see that this is not related to the mass (if we consider no friction with the air).
this goes beyond the ELI5 ideea but we can extract the speed of an object based on a basic mechanics calculation. knowing that energy is always conserved, we determine that the potential energy at a height (h) is equal to the kinetic enegy at the instant before hitting the ground
as such, we have: m * g * h = (m * v ^ 2) / 2
we can divide with the object’s mass (m) and we get: g * h = (v ^ 2) / 2
as we can see, an object’s speed, when reaching the ground is the same, regardless of its weight.
Ok it does effect it but the difference in mass between the two objects makes the gravity from one negligible.
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