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Effective gravity, ge Consider Newton's Second Law on the surface of the earth: In an inertial frame: ΣF = ma = -mgr + other forces (here -r is straight down to the center of the earth) In the rotating frame: ΣF - (2mv*) - (m(r)) - (md*/dtr) = ma* (for the rotating earth, d*/dt = 0) so: other forces - mgr - (2mv*) - (m(r)) = ma* . If we consider the (m(r)) term, we see that for the earth is directed along the axis of the spinning earth, or in the direction of the North star (which at Memphis, is about 35 above the Northern horizon). The direction of r is straight up. Thus the direction of [r] will be {up and to the North} cross {up} gives a direction of {East} with a magnitude that depends on (which is 2π/day), r (which is Rearth) and the sine of the angle between them which is the co-latitude [or cosine of the latitude, which for Memphis would be cos(35)]. Further, cross this vector would be in the direction {North and up} cross {East} = {down and to the North}. The minus sign in front means that the direction of the "fictitious force" would be {up and to the South}. Since this will be essentially constant as long as r remains close to the surface of the earth, we can group this term with the essentially constant -gr to make an effective ge that is a little less than the true g due to gravity and also points a little bit South of straight down to the center of the earth. ω ω×r (East) (E(East) ω×(ω×r) r WRITTEN HOMEWORK PROBLEM #3: a) Calculate the mass of the earth based upon the fact that the Moon orbits the earth with a period of 27 days, 7 hours and 43 minutes at a distance of 384,400 km. (Since the moon orbits in an ellipse rmin = 363,300 km and rmax = 405,500 km, and since the semimajor axis = ½ (rmin + rmax), we get the 384,400 km figure to use. See the example under homework #27 at http://www.cbu.edu/~jholmes/P380/Rrsystem.htm ) b) Using the above mass for the earth and given Rearth = 6,378 km, calculate the magnitude of the true gravitational acceleration, g. (The 6,378 km is the equatorial radius; the polar radius is only 6,357 km.) c) Using the fact that Memphis is close to sea level at latitude 35 North, and given that the period for the earth's spin is 24 hours, calculate the magnitude and direction of the term -((r)) at Memphis. d) Using the results of parts b & c, calculate ge, and compare g and ge.