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Your first job after graduation you are hired by a engineering firm that just got a contract to build the first space elevator. The space elevator consists of a cable hanging from geo-stationary satellite to an anchor point on earth’s surface. Once installed this equipment can be raised and lowered on the elevator, saving the expense and risk associated with rocket launches and reentry. In this problem the radius of the earth will be taken as Re, mass of earth is Me, rotation of the earth (months per day) given by angular frequency is ω (e). The satellite has mass Ms, and system circular orbit of Rs. The elevator cable has length Rs-Re to reach the surface of the earth, and a constant mass per unit length λ.

A) Where are on earth should the anchor station be located in order for the cable to extend vertically upwards from the surface of the earth? And Why?

B) Let the tension on the cable where it attaches to the satellite be T (some variable). What is the Ms of the satellite that the satellite must have to maintain in stable circular orbit? (What mass must the satellite have in order to maintain in stable circular orbit in the presence of the weight of the cable and its own weight?)

C) At what radius Rs in the presence of centripetal force of the weight of the cable in addition to its own weight must the counterweight be? What mass must the satellite have in order to maintain in stable circular orbit in the presence of the weight of the cable and its own weight.

D) What is the tension on the cable as a function of the height? Consider only the weight of the cable and neglect the tension due to external load hanging on the cable.

E) What is the tension at the top of the cable (where it connects to the satellite)? The cable is made of a material with mass density ρ.

F) What’s the minimum force per unit cross sectional area required to maintain its own weight?

G) Both the climbers ascend on the stationary cable at a fixed # of meters of cable per second. What are the radial and tangential components of the velocity of the climber as a function of height? (since the cable is rotating with the earth it has a tangential velocity)

H) What is the force by the cable on the climber? (has to satisfy Newton’s second law)

Do Not Use mg! (Surface of earth diff from orbit)

A given amount of cable will weigh different half way up.

Basically the questions are:

A) Where does the anchor have to be located?

B) What should the mass of the satellite be?

C) What is the minimum theoretical value for the radius of the satellite?

D) What’s the tension of the cable as a function of height?

E) Evaluate that at the top of the cable.

F) How strong does the cable have to be in terms of force per unit area?

G) What’s the velocity of the climber, radial and tangential?

H) What is the force by the cable on the climber?