Craft and Tether
The solution adopted to generate artificial gravity is then to spin the spacecraft by means of thrusters using a discarded rocket stage as a counterweight attached to the crew module by means of a tether.
The center of mass of our spinning spacecraft is found by using the following equation:
where Mc is the mass of the craft for the crew, Rc is the radius of the tether from Mc to the center of mass, Mr is the mass of the counterweight craft, and Rr is the length of the tether from the counterweight to Cm.
Angular speed should be no greater than 2 rpm due to the negative effects caused by motion sickness. Since we want a gravitational pull of 1g in the crew compartment, the length of Rc must be 223.4 meters, as we can see from the following equations where w is the angular speed, R is the radius, and g is the centrifugal acceleration which in this case we want to make equal to the acceleration of gravity.
Substituting 2 rpm for the angular speed (upper limit for acceptable values) we obtain the expected value for the radius (that is, the length of the tether):
R = 223.41 m (4)
It is interesting to see how the radius of rotation varies with the angular speed. Enter a value for the angular speed and then click on the button to find the radius of the tether :
The mass of the crew compartment is determined by operational considerations. The mass of the counterweight craft will also depend on the spacecraft's configuration, on whether it is a discarded stage, etc. Using these numbers in the fisrt equation, we can determine Rr and the full length of tether needed.
Rc = R = 223 m
Ltether = Rc + Rr (5)