Point mass impresicion

[supermap]

While i was wondering i came to wonder how accurate was using point mass gravity for close objects

I immediately stumbled into that... it isnt.

I hope i am doing something wrong but i still dont know what.

So formula for gravitational acceleration is GM/(r^2)

Imagine a pebble 1m above a 4m long block with mass 2

the CoM of the block is 3m below the pebble so acceleration is G(2)/(3^2) = 2G/9 =32G/144

But if you cut the block in half now there are 2 blocks with mass 1

the first CoM is 2m below and the other is 4m... so acceleration is G(1)(2^2) + G(1)(2^2) = 5G/16 = 45G/144

its quite a bigger difference than what i expected... or am i doing something wrong??

[Fractal_UK]

You are correct in that the problem can't be approached in this way. If the closest distance from your pebble to your object is called x1, the point mass approach is only valid when x is approximately equal to x1, i.e. very far away.

The correct way to solve this problem (in 1D) is to split up your long block into infinitesimal slices of a given density and then integrate over the possible values of x. Since your block has mass 2 and length 4, it has 0.5 mass per unit length. The problem you are then solving is the integral GÃÂ/(x^2) between 4 and 2. -0.5G/4 + 0.5G/2 = 1/4G. You will see that this is between your two answers.

(I've changed the terminology here from r to x because using r implies that there is a spherical symmetry that doesn't exist in this system.)

Edited May 15, 2013 by Fractal_UK

[supermap]

Thanks, explained it really clear.

Just left me wondering... when we measure the acceleration on earth... we DO use point mass and we are very close to earth... although we do get the correct result.

Is this because a sphere makes it not to matter or something, because by looking at my first example if we also cut the earth in half, both CoM would move towards and away from the pebble the same ammount... but because distance is squared it would go up once again.

I understand that by using your solution with integration it would work... but when you are asked to calculate the acceleration here on earth you dont care about all that... you just divide the mass by radius squared

[Spacial_Anomaly]

Well, we only can apply the point-mass-approximation for (near)spherically symmetic objects ( http://en.wikipedia.org/wiki/Shell_theorem ). That's why we can assume it in KSP because planets are spherically symmetric.

[K^2]

Point mass approximation only works for approximately spherical objects. For a perfectly spherical object, it is exact.

Edit: Sorry, didn't notice the post right above mine.