Question

A student is given the set of orbital data for some of the moons of Saturn shown below and is asked to use the data to determine the mass Ms of Saturn. Assume the orbits of these moons are circular. Write an algebraic expression for the gravitational force between Saturn and one of its moons. Use your expression from part (a) and the assumption of circular orbits to derive an equation for the orbital period T of a moon as a function of its orbital radius R. Which quantities should be graphed to yield a straight line whose slope could be used to determine Saturn's mass? Complete the data table by calculating the two quantities to be graphed. Label the top of each column, including units. Plot the graph on the axes below. Label the axes with the variables used and appropriate numbers to indicate the scale. Using the graph, calculate a value for the mass of Saturn.

Answer 1

a. F = GM_s m/r^2 b. F_G = F_a GM_s m/r^2 = mv^2/T v = 2 pi r/T Gm_s/r = 4 pi^2 r^2/T^2 T^2 = 4 pi^2 nr^2/6m^2 c. m_s = 4 pi^2/G R^3/T^2 R^3 vs T^2 f. slope = 4.6 - 2.5 times 10^25/4.8 - 2.6 times 10^11 = 2.1 times 10^22/2.2 times 10^10 (4 pi^2/G) = 5.6 times 10^24 kg 9.54 times 10^14.