a satellite of mass m is in a circular orbit with velocity v aroundthe earth (as
ID: 1729638 • Letter: A
Question
a satellite of mass m is in a circular orbit with velocity v aroundthe earth (assumed to have a spherical shape with radius R and massM) denote the universal gravitational constant as G.a. Assuming that the centripetal force acting on the satellite isprovided by the gravitational force due to the earth with its masslocated at the center, and its radius is equal to R, that isneglecting the height of the satellite above the earth's surface ascompared to R, obtain an expression for the velocity v of thesatellite in terms of G, M, and R.
b. Using G=6.674 * 10^-11 N.m^2/kg^2 M= 5.98 * 10^24kg R=6.38* 10^6m Mass of satellite m=1.00 * 10^5kg, calculate thevalueof the velocity v and the centripetal force acting on thesatellite.
Explanation / Answer
a. Okay, so first let's think about the problem for a sec. We don'twant the satellite to fall to earth, so we're going to balance theforce of gravity with the centripetal force. We can only get thecentripetal force when moving rapidly in a circle, so we'rebasically going to spin the satellite around the planet at exactlythe right speed so it's not falling or rising above the surface ofthe planet. Definitions: G = Universal gravitational constant M = Mass of planet m = Mass of satellite R = Radius of orbit v = Velocity of satellite (tangential) We'll say that towards the planet (inwards) is positive and awayfrom the planet (outwards) is negative. And two equations: Fg = Force of Gravity =GMm/R2Fc = Centripetal Force =mv2/R The sum of all forces must equal zero so we have no acceleration.Don't worry about the "F" it's just a fancy way of saying"the sum of forces." F = GMm/R2 - mv2/R = 0 Some algebra 1. GMm/R2 = mv2/R (Multiply by R) 2. GMm/R = mv2 (Divide by m) 3. GM/R = v2 (Take the square root of both sides andrearrange) 4. v = (GM/R) There's the answer to a! b. Simply plug in numbers. v = (GM/R) = (6.674e-11*5.98e34/6.38e6)0.5 =790921450 m/s = 7.91e8 m/s Fc = mv2/R = 1.00e5*790921450/6.38e6 =12396887 N = 1.24e7 N Remember you have three significant figures going in, so you canonly determine your answer to three significant figures. Cheers!
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