ps04sol

# ps04sol - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department...

This preview shows pages 1–4. Sign up to view the full content.

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,

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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)
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?

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## 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.

### Page1 / 17

ps04sol - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online