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Laws of gravitation

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I have been confused for some while by what seems to be a contradiction between the principles of gravitation expounded by Galileo and those expounded by Newton - both of which we know to be correct, so I must be missing something here! Galileo established that two objects - regardless of differences in size, mass or shape - will, discounting air resistance , fall to the ground at the same rate. We had this principle impressively demonstrated by astronaut Dave Scott during his moon walk on the Apollo 15 mission when, taking advantage of the absence of any atmosphere on the moon, he simultaneously dropped a feather and a hammer onto the moon's surface.

On the other hand Newton's formulation of the force of gravitational attraction between any two objects states that this is proportional to the product of the mass of BOTH objects. SO it seems to me that if the strength of gravitational attraction between the ground and a hammer on the one hand and of a feather on the other, depends at least partly on the mass of those two objects, as well as the mass of the earth (or the moon) then the calculation of those forces ought to produce very different figures in each case. 

Where am I going wrong here?

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Yes, the force is different.  But what you see when you drop something is the effect of acceleration, and

f = m * a

So

a = f / m

The force is higher, but the mass is too, so the acceleration (hence rate of falling) is the same.

 

 

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Well this is how I see it using the two basic formula f = m * a and f = (G x M x m)/ (r^2) where in this case M is the mass of the Earth, m is the mass of a feather, or a man or an elephant, and r is the mean equatorial radius of the Earth.

IMG_8484.thumb.jpg.a88fa047434c01d5d46f892d051c292e.jpg

So all objects fall at the same rate, i.e. acceleration due to gravity - neglecting air resistance.

Does that help - or confuse?

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The significance of this observation, that inertial and gravitational mass (the two m's in the two equations) are apparently exactly equal and why this should be so, runs very deep and was finally neatly solved by Einstein's theory of general relativity. ie that gravity is not actually  a force but a curvature of spacetime caused by the masses.

Edited by robin_astro
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I assume it's like, 'the speed  of light in a vaccum is slightly higher.' Yes, you can measure the speed of light in a vacuum, but it's not relevant to do an experiment and expect it to be so in an uncontrolled environment. Yes, with no resistance objects will fall - all of them. 

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