It’s pretty long, so I really, really appreciate yourhelp in any or all of these
ID: 1749723 • Letter: I
Question
It’s pretty long, so I really, really appreciate yourhelp in any or all of these question. I have a test tomorrow soplease solve this so I can learn from it. I have no clue.Here is the question :
In your first job after graduation, you are hired by anengineering firm that just won a contract to design the first space elevator. A space elevator consistof a cable hanging from a geostationary satellite to an anchor point on the earth'ssurface. Once its installed, goods an equipment can be raised and lowered on the elevator, saving the expenseand risk associated with rocket launches and re-entry. In thisproblem, the radius of the earth will be taken by Re, and the massof the earth as Me and the rotation of the earth--months perday--given by the angular frequency . The satellite hasa mass Ms and it sits in an circular orbit at radius Rs. Thesatellite is also know as counterweight. The elevator cable has a length (Rs - Re) and it has aconstant mass per unit length lamda.
1) Where on earth should the anchor station be located inorder for the cable to ascend vertically up from the surface of theearth? Why? [ I am guessing at the earth's equator because atthat point it is the most stationary to geosynchronous orbit, meaningthe axis is not tilting much.]
2) Let the tension of the cable where it attaches to thesatellite be Ts. What is the mass Ms of the satellite, that thesatellite must have in order to remain in stable circular orbit atradius Rs in presence of the centripetal force . Inother words, what mass does the satellite must have in order tomaintain in stable circular orbit in presence of the weight of thecable and its own weight?
3) What is the minimum theoretical value, Xr, for the radiusof the satellite, that’s possible for any satellite massMx.
4) What is the tension on the cable as a function of theheight above the surface of the earth. Consider only the weight ofthe cable and neglect the tension due to external loads, hanging onthe cable.
5) What is the tension at the top of the cable (right where ithooks to the top of the satellite) ?
6) The cable is made up with a materialmass-density, . What’s the minimum force pera unit cross section areas required for the cable to sustained itsown weight?
7) Suppose a climber ascends the stationary cable at a fixednumber of meters of cable per second. What are the radial and thetangential components of the velocity of the climber as a functionof the height.
8) What is the force by the cable on the ascender (theclimber).
HINT : Do not use mg. G varies. So a given an amount of cablewill weigh a different amount halfway up then it does on thesurface on the earth. So it requires an integral. To get the tension you need to compute the weight of all thepieces below you and each one of the will have a different weightper unit mass, depending on its height.
It’s pretty long, so I really, really appreciate yourhelp in any or all of these question. I have a test tomorrow soplease solve this so I can learn from it. I have no clue.
Here is the question :
In your first job after graduation, you are hired by anengineering firm that just won a contract to design the first space elevator. A space elevator consistof a cable hanging from a geostationary satellite to an anchor point on the earth'ssurface. Once its installed, goods an equipment can be raised and lowered on the elevator, saving the expenseand risk associated with rocket launches and re-entry. In thisproblem, the radius of the earth will be taken by Re, and the massof the earth as Me and the rotation of the earth--months perday--given by the angular frequency . The satellite hasa mass Ms and it sits in an circular orbit at radius Rs. Thesatellite is also know as counterweight. The elevator cable has a length (Rs - Re) and it has aconstant mass per unit length lamda.
1) Where on earth should the anchor station be located inorder for the cable to ascend vertically up from the surface of theearth? Why? [ I am guessing at the earth's equator because atthat point it is the most stationary to geosynchronous orbit, meaningthe axis is not tilting much.]
2) Let the tension of the cable where it attaches to thesatellite be Ts. What is the mass Ms of the satellite, that thesatellite must have in order to remain in stable circular orbit atradius Rs in presence of the centripetal force . Inother words, what mass does the satellite must have in order tomaintain in stable circular orbit in presence of the weight of thecable and its own weight?
3) What is the minimum theoretical value, Xr, for the radiusof the satellite, that’s possible for any satellite massMx.
4) What is the tension on the cable as a function of theheight above the surface of the earth. Consider only the weight ofthe cable and neglect the tension due to external loads, hanging onthe cable.
5) What is the tension at the top of the cable (right where ithooks to the top of the satellite) ?
6) The cable is made up with a materialmass-density, . What’s the minimum force pera unit cross section areas required for the cable to sustained itsown weight?
7) Suppose a climber ascends the stationary cable at a fixednumber of meters of cable per second. What are the radial and thetangential components of the velocity of the climber as a functionof the height.
8) What is the force by the cable on the ascender (theclimber).
HINT : Do not use mg. G varies. So a given an amount of cablewill weigh a different amount halfway up then it does on thesurface on the earth. So it requires an integral. To get the tension you need to compute the weight of all thepieces below you and each one of the will have a different weightper unit mass, depending on its height.
It’s pretty long, so I really, really appreciate yourhelp in any or all of these question. I have a test tomorrow soplease solve this so I can learn from it. I have no clue.
Here is the question :
In your first job after graduation, you are hired by anengineering firm that just won a contract to design the first space elevator. A space elevator consistof a cable hanging from a geostationary satellite to an anchor point on the earth'ssurface. Once its installed, goods an equipment can be raised and lowered on the elevator, saving the expenseand risk associated with rocket launches and re-entry. In thisproblem, the radius of the earth will be taken by Re, and the massof the earth as Me and the rotation of the earth--months perday--given by the angular frequency . The satellite hasa mass Ms and it sits in an circular orbit at radius Rs. Thesatellite is also know as counterweight. The elevator cable has a length (Rs - Re) and it has aconstant mass per unit length lamda.
1) Where on earth should the anchor station be located inorder for the cable to ascend vertically up from the surface of theearth? Why? [ I am guessing at the earth's equator because atthat point it is the most stationary to geosynchronous orbit, meaningthe axis is not tilting much.]
2) Let the tension of the cable where it attaches to thesatellite be Ts. What is the mass Ms of the satellite, that thesatellite must have in order to remain in stable circular orbit atradius Rs in presence of the centripetal force . Inother words, what mass does the satellite must have in order tomaintain in stable circular orbit in presence of the weight of thecable and its own weight?
3) What is the minimum theoretical value, Xr, for the radiusof the satellite, that’s possible for any satellite massMx.
4) What is the tension on the cable as a function of theheight above the surface of the earth. Consider only the weight ofthe cable and neglect the tension due to external loads, hanging onthe cable.
5) What is the tension at the top of the cable (right where ithooks to the top of the satellite) ?
6) The cable is made up with a materialmass-density, . What’s the minimum force pera unit cross section areas required for the cable to sustained itsown weight?
7) Suppose a climber ascends the stationary cable at a fixednumber of meters of cable per second. What are the radial and thetangential components of the velocity of the climber as a functionof the height.
8) What is the force by the cable on the ascender (theclimber).
HINT : Do not use mg. G varies. So a given an amount of cablewill weigh a different amount halfway up then it does on thesurface on the earth. So it requires an integral. To get the tension you need to compute the weight of all thepieces below you and each one of the will have a different weightper unit mass, depending on its height.
It’s pretty long, so I really, really appreciate yourhelp in any or all of these question. I have a test tomorrow soplease solve this so I can learn from it. I have no clue.
Here is the question :
In your first job after graduation, you are hired by anengineering firm that just won a contract to design the first space elevator. A space elevator consistof a cable hanging from a geostationary satellite to an anchor point on the earth'ssurface. Once its installed, goods an equipment can be raised and lowered on the elevator, saving the expenseand risk associated with rocket launches and re-entry. In thisproblem, the radius of the earth will be taken by Re, and the massof the earth as Me and the rotation of the earth--months perday--given by the angular frequency . The satellite hasa mass Ms and it sits in an circular orbit at radius Rs. Thesatellite is also know as counterweight. The elevator cable has a length (Rs - Re) and it has aconstant mass per unit length lamda.
1) Where on earth should the anchor station be located inorder for the cable to ascend vertically up from the surface of theearth? Why? [ I am guessing at the earth's equator because atthat point it is the most stationary to geosynchronous orbit, meaningthe axis is not tilting much.]
2) Let the tension of the cable where it attaches to thesatellite be Ts. What is the mass Ms of the satellite, that thesatellite must have in order to remain in stable circular orbit atradius Rs in presence of the centripetal force . Inother words, what mass does the satellite must have in order tomaintain in stable circular orbit in presence of the weight of thecable and its own weight?
3) What is the minimum theoretical value, Xr, for the radiusof the satellite, that’s possible for any satellite massMx.
4) What is the tension on the cable as a function of theheight above the surface of the earth. Consider only the weight ofthe cable and neglect the tension due to external loads, hanging onthe cable.
5) What is the tension at the top of the cable (right where ithooks to the top of the satellite) ?
6) The cable is made up with a materialmass-density, . What’s the minimum force pera unit cross section areas required for the cable to sustained itsown weight?
7) Suppose a climber ascends the stationary cable at a fixednumber of meters of cable per second. What are the radial and thetangential components of the velocity of the climber as a functionof the height.
8) What is the force by the cable on the ascender (theclimber).
HINT : Do not use mg. G varies. So a given an amount of cablewill weigh a different amount halfway up then it does on thesurface on the earth. So it requires an integral. To get the tension you need to compute the weight of all thepieces below you and each one of the will have a different weightper unit mass, depending on its height.
Explanation / Answer
2) Let the tension of the cable where it attaches to thesatellite be Ts. What is the mass Ms of the satellite, that thesatellite must have in order to remain in stable circular orbit atradius Rs in presence of the centripetal force . Inother words, what mass does the satellite must have in order tomaintain in stable circular orbit in presence of the weight of thecable and its own weight?Total force on satellite = m v2 / R . G M m /Rs2 + Ts = m2 R . mass of satellite then: m = TsRs2 / (2Rs3 + G Me) . 3) What is the minimum theoretical value, Xr, for the radiusof the satellite, that’s possible for any satellite massMx. this question doesnt make much sense... but I think youreasking for the minimum possible value of Rs . Thiswould be simply when the tension was zero, so: . G M m /Rs2 + 0 = m2 R . Rmin = ( G M / 2)1/3 . 4) What is the tension on the cable as a function of theheight above the surface of the earth. Consider only the weight ofthe cable and neglect the tension due to external loads, hanging onthe cable. . The tension decreases in the cable as rincreases. You can write the difference in tension in the cableover a very small distance dr as: . dT - G M dm / r2 = - dm2 r because asmall segment of the cable, with mass dm, experiencescentripetal acceleration due to the tension in the cable fromeither side (i.e. Tension below - Tension above) andthe gravitational force pulling inward. Note that the + direction is defined as away from the center of the Earth(i.e. radial direction). . Also... dm = dr so . dT = - dm (GM r-2 + 2 r) . dT = - dr (G M r-2 + 2 r) . To determine tension as a function of r, just integrate bothsides. You can integrate the left side from 0 toT and the right side from Rs to (h + Re ) . Then you have a big algebraic mess, but it will be T as a function of h (i.e. height above theearths surface). . 5) What is the tension at the top of the cable (right where ithooks to the top of the satellite) ? . According to the given info in part 4, the tension atthe top of the cable is zero. Because its not connected toanything. . 6) The cable is made up with a materialmass-density, . What’s the minimum force per aunit cross section areas required for the cable to sustained itsown weight? . Get the tension at h = 0 (use your result frompart 4). This is the maximum tension on the cable (at the base,i.e. the ground). Then you can write: . density * area * length = mass = length * . area = / . So the minimum strength of the material mustmatch the maximum stress it will be subjected to. So... . minimum force / area required = tension at h=0 / area = tension at h=0 / (/) or . * tension at h=0 / .
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