ps04sol - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department...

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Physics 8.01 Problem Set 4: Circular Motion Dynamics Solutions Problem 1: A geostationary satellite goes around the earth once every 23 hours 56 minutes and 4 seconds, (a sidereal day, shorter than the noon-to-noon solar day of 24 hours) so that its position appears stationary with respect to a ground station. The mass of the earth is m e = 5.98 ! 10 24 kg . The mean radius of the earth is 6 e 6.37 10 m R = ! . The universal constant of gravitation is G = 6.67 ! 10 " 11 N # m 2 # kg " 2 . Your goal is to find the radius of the orbit of a geostationary satellite. Describe what motion models this problem. What is the radius of the orbit of a geostationary satellite? Approximately how many earth radii is this distance? Solution: The satellite’s motion can be modeled as uniform circular motion. The gravitational force between the earth and the satellite keeps the satellite moving in a circle. The acceleration of the satellite is directed towards the center of the circle, that is, along the radially inward direction. The figure below is close to a scale drawing. Choose the origin at the center of the earth, and the unit vector ˆ r along the radial direction. This choice of coordinates makes sense in this problem since the direction of acceleration is along the radial direction. Let r ! be the position vector of the satellite. The magnitude of r ! (we denote it as s r ) is the distance of the satellite from the center of the earth, and hence the radius of its circular orbit. Let ! be the angular velocity of the satellite, and the period is T = 2 / " . The acceleration is directed inward, with magnitude 2 s r ; in vector form,
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2 s ˆ r ! = " a r ! . (1.1) Apply Newton’s Second Law to the satellite for the radial component. The only force in this direction is the gravitational force due to the Earth, 2 grav s s ˆ F m r = " r ! . (1.2) The inward radial force on the satellite is the gravitational attraction of the earth, 2 s e s s 2 s ˆ ˆ m m G m r r " = " r r . (1.3) Equating the ˆ r components, 2 s e s s 2 s m m G m r r = . (1.4) Solving for the radius of orbit of the satellite r s , 1/3 e s 2 Gm r " # = $ % & ' . (1.5) The period T of the satellite’s orbit in seconds is 86164 s and so the angular speed is 5 1 2 2 7.2921 10 s 86164 s T " # # = = = $ . (1.6) Using the values of e , and G m in Equation (1.5), we determine s r ; 7 s e 4.22 10 m 6.62 r R = ! = . (1.7)
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Problem 2: Mo swings a ball of mass m in a circle of radius R in a vertical plane by means of a massless string. The speed of the ball is constant and it makes one revolution every t 0 seconds. a) Find an expression for the radial component of the tension in the string T ( ! ) as a function of the angle the ball makes with the vertical 1 . Express your answer in terms of some combination of the parameters m , R , t 0 and the gravitational constant g . b) Is there a range of values of t 0 for which this type of circular motion can not be maintained? If so, what is that range?
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This note was uploaded on 04/13/2011 for the course PHYSICS 8.01 taught by Professor Guth during the Fall '09 term at MIT.

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ps04sol - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department...

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