An Earth satellite in a circular orbit of radius R has a period T. What is the period of an Earth satellite in a circular orbit of radius 4R ?

Question

BlackTom AI · Accepted Answer

We start from Kepler's third law for orbital motion around a central body like Earth: the orbital period squared is proportional to the semi-major axis cubed, i.e., T^2 ∝ r^3 for circular orbits. Let the initial orbit have radius R and period T, so T^2 ∝ R^3. For a second orbit with radius 4R, its period T' must satisfy T'^2 ∝ (4R)^3 = 64 R^3. Taking the ratio of the two equations: (T')^2 / T^2 = (64 R^3) / (R^3) = 64. Therefore T' = T * sqrt(64) = T * 8. In words, when the orbital radius increases by a factor of 4, the period increases by a factor of 8, due to the cubic dependence of radius in Kepler's law. This aligns with the general relation T ∝ r^(3/2) for circular orbits around a given central mass. If one were to check the dimensional consistency, time scales with the radius to the 3/2 power, so multiplying the radius by 4 multiplies the period by 4^(3/2) = 8. Consequently, the period for the 4R orbit is eight times the original period T.

An Earth satellite in a circular orbit of radius R has a period T. What is the period of an Earth satellite in a circular orbit of radius 4R ?

View Explanation

Log in for full answers

Similar Questions

By using either the Doppler or Transit methods, you can determine the orbital period of the planet. Using just that information, what else can you determine about the planet? (Hint: think Kepler's laws).

A planet in our solar system whose orbital semi-major axis is larger than that of the Earth's will have an orbital period, compared to the Earth, which is

Kepler's third law states that the square of the orbital period of a planet is proportional to

How do Astronomers measure the mass of the Sun (about 2x1030kg!)?

Kepler's third law, means that (select the most correct answer)

Imagine we've discovered a planet orbiting another star at 1 AU every 6 months. The planet has a moon that orbits the planet at the same distance as our Moon, but it takes 2 months. What can we infer about this planet? (Hint: Use Newton's Version of Kepler's Third Law)

In a consumer society, many adults channel creativity into buying things

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler