2.11 This problem shows how the gravitational force can be a centripetal force a

Question

2.11 This problem shows how the gravitational force can be a centripetal force a) The latitude of Memphis is approximately 35°. Find the tangential speed of our classroom due to the spinning of the Earth. The Earth's radius is 6380 km. b) Find the tangential speed in m/s that we would be traveling if our classroom were located on Jarvis Island. Jarvis Island is approximately on the Equator in the South Pacific Ocean. c) Find the angular speed in rad/sec of our Jarvis Island classroom. d) Starting with Newton's Second Law, found how high above the Earth's surface a satellite must be so that it is constantly over our Jarvis Island classroom. Use 5.98x1024 kg for the Earth's mass. (Such a satellite is in a "geostationary" orbit.) e) What is the tangential speed in m/s of such a satellite? * Geostationary satellites are routinely used for weather tracking, geological surveying, communications, and other applications. Our problem is simplified somewhat since we assume that the Earth is a perfect sphere, we have located our classroom on the equator, and we assume a perfect circular orbit.

Explanation / Answer

2.11 a)

r = radius = 6380 km = 6.38 x 106 m

T = time period of earth = 24 h = 24 x 3600 sec = 86400 sec

w = angular speed = 2pi/T

linear tangential speed is given as

v = r w Cos35

v = 2pir Cos35/T = 2 (3.14) (6.38 x 106) Cos35 /(86400) = 380 m/s

b)

linear tangential speed at equator is given as

v = r w

v = 2pir /T = 2 (3.14) (6.38 x 106) /(86400) = 464 m/s

c)

w = angular speed = 2pi/T = 2 (3.14)/(86400) = 7.27 x 10-5 rad./s

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