On Written Homework #1 - Dimensional Analysis, how would we solve for questions from 1b to 2?

l. [15 pts} Solar System Systematics. The gravitational force between two objects ofmass m, ":1\": y and my which are separated by a distance r is equal to F = G . The dimensions of force are [M] [L]![T]2. a. [2 pts} What are the dimensions of Newton 's Constant G? Show your work. El i.- I" h. [4 pts} Let's apply this law to the motion of the planets in the Solar System and approximate the motion of the planets as perfect circles :1er the Sun. Soon we will learn Newton's Second Law of Motion, which says F : ma , and that the acceleration of an object ht circular motion is related to the radius of the circle, R, and the time i" it takes to go around the circle as n = 433R? T2. Armed with these facts, if we were to take a planet and double its distance from the Sun, by what numerical factor would the planet's \"year\" {the time it takes to orbit the Sun} change? Show your work. c. [3 pts} From values that you can fntd in re textbook for the orbit of Mars, what numerical value of GM,\" do you infer? Show your work. Important: Do not use the values that you can nd separately for G or M5,\". d. [2 pts} Consider te Earth and its Moon as its own \"solar system" wit the Earth at the center. From the Moon's distance and orbital period, what is GMEm? {Historical Note: Newton did not know the value of Newton's Constant, nor did he know the masses of either the Sun or Earth, but he did know the values of EMS", and SHE-.) e. [2 pts} According to your answer in part d, at what height above the Earth would a satellite need to be for it to take 24 hours to orbit the Earth? Show your work. 1'. [2 pts} From your result in part d, what is the acceleration of the International Space Station {ISS} which is in orbit EDD2513 miles above the surface of the Earth? 2. [5 pts} Simple Pendulum. What is the dimensional analysis prediction for the period of a simple pendulum? A simple pendulum consists of a single mass m on the end of a massless string or stick of length L that hangs from and eet y pivots about a support point at its other end. The \"period' of the pendulum means the total time it takes to sw ing back and forth once. As always, show your work