Question

To teach you how to find the parameters characterizing an object in a circular orbit around a much heavier body like the earth. The motivation for Isaac Newton to discover his laws of motion was to explain the properties of planetary orbits that were observed by Tycho Brahe and analyzed by Johannes Kepler. A good starting point for understanding this (as well as the speed of the space shuttle and the height of geostationary satellites) is the simplest orbit--a circular one. This problem concerns the properties of circular orbits for a satellite orbiting a planet of mass M. For all parts of this problem, where appropriate, use G for the universal gravitational constant. Part A) Find the orbital speed v for a satellite in a circular orbit of radius R. (Express the orbital speed in terms of G, M, and R),Part B) Find the kinetric energy K of a satellite with mass m in a circular orbit with radius R. (Express your answer in terms of m, M, G, and R).Part D) Find the orbital period T. (Express your answer in terms of G, M, R, and pi).

Answer 1

Answer:

a) v = √(GM/R), b)   K = ½ G m M /R and d)  T = 2π R√(R/GM)

Explanation:

This problem should use the law of universal gravitation

    F = G m M / r²

a) For this part we use Newton's second law where acceleration is centripetal

    F = m a

Centripetal acceleration is

    a = v² / r

    F = m v² / r

    G m M / r² = m v² / r

    G M / r = v²

We use the distance (R) measured from the center of the planet

    v = √(GM / R)

b) the expression for kinetic energy is

    K = ½ m v²

    K = ½ m G M / R

    K = ½ G m M / R

d) as the velocity module is constant, we can use the equation and uniform motion

    v = d / T

    T = d / v

The distance is the length of the circle

   d = 2π R

   T = 2π R / √(GM / R)

   T = 2π R √ (R / GM)