Artificial gravity: To compensate for weightlessness in a space station, artificial gravity can be produced by rotating the station.8 The required number N of rotations per minute is a function of two variables: the distance r to the center of rotation, and a, the desired acceleration (or magnitude of artificial gravity). See Figure 2.58 on the following page. The formula is
We measure r in meters and a in meters per second per second.
a. First we assume that we want to simulate the gravity of Earth, so a = 9.8 meters per second per second.
i. Find a formula for the required number N of rotations per minute as a function of the distance r to the center of rotation.
ii. Make a graph of N versus r. Include distances from 10 to 200 meters.
iii. What happens to the required number of rotations per minute as the distance increases? Explain your answer in practical terms.
iv. What number of rotations per minute is necessary to produce Earth gravity if the distance to the center is 150 meters?
b. Now we assume that the distance to the center is 150 meters (so r = 150).
i. Find a formula for the required number N of rotations per minute as a function of the desired acceleration a.
ii. Make a graph of N versus a. Include values of a from 2.45 (one-quarter of Earth gravity) to 9.8 meters per second per second.
iii. What happens to the required number of rotations per minute as the desired acceleration increases?