We use Kepler's third law to find the correct answer is:
- The period (year) of the plant increases with increasing distance to the Sun
- The period is independent of the mass of the Planet
- The period is inversely proportional to the masses of the Sun
Kepler's third law is a direct application of second Newton's law the motion of the planets around the Sun, if we assume that the orbit is circular the deduction is
F = m a
where F is the force of attraction between the Sun and the planet, m is the mass of the planet and a is the ccentripetal acceleration.
The force is given by universal gravitation law, which says that the force of attraction between two bodies is directly proportional to the mass of the bodies and inversely proportional to the square of the distance between them, this force is attractive
F = [tex]G \frac{M \ m}{r^2}[/tex]
where G is the universal constant of attraction, M the mass of the Sun, m the mass of the plant and r the distance between them
From kinematics we know the expression of the centripetal acceleration
a = [tex]\frac{v^2}{r}[/tex]
we substitute
[tex]G \frac{M \ m}{r^2 } = m \frac{v^2}{r}\\G \frac{M}{r} = v^2 \\[/tex]
As we assumed a circular orbit, the velocity of the planet in the orbit is constant, so the kinematic relations of uniform motion can be used.
The distance traveled by the planet is the length of the circle
d = 2π r
v = d / T
where T is the period of the orbit
v = 2π r / T
we substitute
[tex]G \frac{M}{r} = \frac{4 \pi ^2 r^2 }{t^2}\\T^2 = ( \frac{4 \pi ^2}{G \ M} ) \ r^3[/tex]
The year is the duration of the period of the planet's orbit around the Sun, with the previous expression it is observed that the period (year) increases when the radius of the orbit increases with a power of 1.5, in summary the planets more far away has duration of the greater anus
The constant in parentheses is inversely proportional to the greater of the Sun, that is, for more massive stars the period decreases
The period is in dependent on the mass of the planet, therefore any object at the same distance has the same duration of the period or the year of the planet.
In conclusion we use Kepler's third law to find the correct answer is:
- The period (year) of the plant increases with increasing distance to the Sun
- The period is independent of the mass of the Planet
- The period is inversely proportional to the masses of the Sun.
Learn more about Kepler's third law here:
https://brainly.com/question/1819012
