posted by Lena
Why is acceleration the same for all free-falling objects regardless of mass and distance? Please don't say according to this formula or something. could you please actually explain the theroy behind it.
It is not the same regardless of distance. As one moves away from the surface of the earth, the acceleration is less.
The idea here is in Potential Theory, and I remember Kelloggs massive Dissertation on it.http://books.google.com/books?id=TxlfQi46CvEC&dq=kelloggs+potential+theory&printsec=frontcover&source=bl&ots=sxjfNE5pOX&sig=rMCYmWeiEwczNzjSL3P3oXfk6vo&hl=en&ei=E8_USaDYO6LwswO6k9GkCg&sa=X&oi=book_result&ct=result&resnum=1#PPP9,M1
And Newton wrote on his Potential Theory also, but the writing is not easily clear.
The Theory of Potentials lays out the potential field for matter, and in gravity, one gets the potential function as a function of 1/distance. So when one leaves Earth, the potential of Gravity decays as the reciprocal of distance from the center of Earth. Thus, when one is two Earth radii from Earth, the potential has decreased by 1/2
Now it is this potential which causes the gravitational field to vary, and of course the gravitational field causes forces, which induce motion (acceleration). The potential function is equivalent to stored energy in space, and it is equal over small increments in space as force*changesindistance.
given this, then we have a known (ignoring the calculus involved)
force is proportional to 1/distance^2
and thus, acceleration is proportional to force (Newtons second law).
Now if one goes great distances from Earth, for instance in high altitude orbits (2000km), the difference in acceleration due to gravity is easily measured. But changes on the surface of Earth itself (sea level vs an altitude of 300 m) is not easily deteceted, as the change in distance from the EArth center is trival.
So if I dropped the same ball twice at different heights suppose 1m and 500m the ball that was dropped at 500m would have a lower acceleratino but greater velocity?
2. A square of side is removed from one corner of a square sandwich that has sides of length L. The center of mass of the remainder of the sandwich moves from C to C’. The displacement of the x coordinate of the center of mass (from C to C’) is
1/12 * L