9.2. Newton’s Universal Law of Gravity http://www.ck12.org
A natural question at this point is- How did Newton arrive at this equation describing the forces between objects? No
doubt, Newton was a genius, but even a genius seeks reassurance. We will see that Kepler’s work helped to convince
Newton that his final form for the Universal Law of Gravity was quite reasonable.
http://www.youtube.com/watch?v=391txUI76gM
Newton’s Verification of His Inverse Square Law
In order to understand why Newton believed in the validity of the Universal Law of Gravity, we need to go back
to an earlier lesson in which we learned that the acceleration of an object in circular motion is directed toward the
center of the circle it travels in and has acceleration,a, of magnitudea=v
2
r.We will assume the orbit of the moon is circular in the example below (it is really slightly elliptical), that the distance
between the Earth and Moon from center to center isR, that the magnitude of the Moon’s velocity isv, and that the
period of the Moon’s orbit about the earth isT. Recall that the circumference of a circle isC= 2 πR. SeeFigure9.7.
FIGURE 9.7
The Moon in orbit about the Earth with
velocity v.Sincex=vt, we can substituteC, the distance the moon travels during one orbit about the Earth, forx, and substitute
2 πRforCThus,x=vt→C=vT→ 2 πR=vTsolving forv→v=^2 πTR.
If we now replacevwith^2 πTRina=v
2
randrwithR, we have,a=
( 2 πR)^2
T^2
R →4 π^2 R
T^2 , thusa=4 π^2 R
T^2. This is the centripetal
acceleration of the Moon relative to the Earth. The force that the Earth exerts upon the Moon, according to Newton’s
Second Law, is∑F=mma=mm^4 π
(^2) R
T^2 , (Equation 1), wheremmis the Moon’s mass.
On the other hand, according to the Universal Law of Gravity, theFin Equation 1 (seeFigure9.7) is the same as
theFas inF=GmRm 2 me, Equation 2, wheremeis the mass of the Earth.
Thus, we can equate the two forces: