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

Answer: B. Can be used to derive Kepler's third law of planetary motion.

Explanation:

According to Newton's law of universal gravitation:

$F=G\frac{Mm}{a^2}$   (1)

Where:

$F$ is the module of the force exerted between both bodies

$G$ is the universal gravitation constant.

$M$ and $m$ are the masses of both bodies.

$a$ is the distance between both bodies

If we apply this law for two bodies in a circular orbit:

$G\frac{Mm}{a^2}=m{\omega}^{2}a$   (2)

Where:

$\omega=\frac{2\pi}{T}$ is the angular velocity, which is related to the period of the orbit $T$

Substituting $\omega$ in (2):

$G\frac{Mm}{a^2}=m{(\frac{2\pi}{T})}^{2}a$   (3)

Finding $T^{2}$:

$T^{2}=\frac{4\pi^{2}}{GM}a^{3}$   (4)

Knowing $\frac{4\pi^{2}}{GM}=C$  is a constant:

$T^{2}=Ca^{3}$  

We finally get to 3rd Kepler's law of planetary motion:

$\frac{T^{2}}{a^{3}}=C$   (5)