Question

Newton’s law of universal gravitation.

a.
is equivalent to Kepler’s first law of planetary motion.
b.
can be used to derive Kepler’s third law of planetary motion.
c.
can be used to disprove Kepler’s laws of planetary motion.
d.
does not apply to Kepler’s laws of planetary motion.

Answer 1

@Mashy Help please?

Answer 2

Answer's B.

Answer 3

When you combine Newton's gravitation and circular acceleration, which must balance in order for the object to remain in orbit, you get a nice relation between the period, distance, and mass of the central body. It beings by equating the centripetal force (Fcent) due to the circular motion to the gravitational force (Fgrav): Fgrav = Fcent

Fgrav = G m1 m2 / r2
Fcent = m2 V2 /r

Let the Earth be m1 and the Moon be m2. For circular motion the distance r is the semi-major axis a. The orbital velocity of the Moon can be described as distance/time, or circumference of the circular orbit divided by the orbital period:

V = 2 pi r /P

so setting the forces equal yields

G m1 m2 / a2 = m2 V2 /a

note that the m2 will cancel, so that circular orbital motion is independent of the mass of the orbiting body!

G m1 / a2 = ((2 pi a)2/P2)/a

which we rearrange to place all the a-terms on the right and all the P-terms on the left:

G m1/(4 pi2) P2 = a3

Answer 4

\[\frac{ Gm _{1} }{4 \pi ^2}p^2 = a ^3\]

Answer 5

which should look startlingly like Kepler's third law

Answer 6

Thank you Dexter!

Answer 7

You're welcome :)