keplers third law

Question

Kepler's Third Law of planetary motion states that the square of the period T of a planet (the time it takes for the planet to make a complete revolution about the sun) is directly proportional to the cube of its average distance d from the sun.

(a) Express Kepler's Third Law as an equation. (Use k for the constant of proportionality.)

 

(b) Find the constant of proportionality by using the fact that for our planet the period is about 365 days and the average distance is about 93 million miles. (Enter your answers in scientific notation and round the decimal value to two decimal places)

k

(c) The planet Neptune is about 2.79 × 10⁹ mi from the sun. Find the period of Neptune. (Round your answer to the nearest whole number of years.)

Answer 1

a. T² = kd³

b. (365)² = k (93000000)³

k = (365)² / (93000000)³ = 1.66 x 10^-19

c. T² = (1.66 x 10^-19)(2.79 x 10⁹)³ = 4.6314 x 10⁸

So, T = 2.15 x 10⁴ days (or about 58.90 Earth years)