A space habitat for a long space voyage consists of two cabins each connected by

Question

A space habitat for a long space voyage consists of two cabins each connected by a cable to a central hub. The cabins are set spinning around the hub axis, which is connected to the rest of the spacecraft to generate artificial gravity.

(a) What forces are acting on an astronaut in one of the cabins?

(b) Write Newton's second law for an astronaut lying on the "floor" of one of the habitats, relating the astronaut's mass m, his velocity v, his radial distance from the hub r, and the normal force n.
Newton's 2nd Law= ?
Normal Force= ?

(c) What would n have to equal if the 65.0-kg astronaut is to experience half his normal Earth weight?

(d) Calculate the necessary tangential speed of the habitat from Newton's second law.

(e) Calculate the angular speed from the tangential speed.

(f) Calculate the period of rotation from the angular speed.

(g) Calculate the tangential speed at the top of his head if he is 2.08 m tall.

Explanation / Answer

(a) Since there must be a force that keeps the astronaut in the rotating circle, we know that the net force is towards the center of the circle. Since there is no gravity, the only force must be the centripetal force (Normal force) towards the center of the hub. (b) Newton's second law: mv^2/r normal force: mv^2/r (c) Normal force=ma Normal force=(65)(9.8)= 637 N Half Normal Force = 637/2 = 318.5 (d) F = mv^2/r 318.5 =(65)(v^2)/10 3185 =(65)(v^2) v = 7m/s (e) w = v/r = 7/10 = 0.7 rad/s (f) Period = 2 pi / w = 2 * 3.14 /0.7 =9.976 s (g) The astronaut's head will be moving faster than his feet because the radius from his head to the center of the hub is shorter than that from his feet to the hub. (i) v = (r-h)w =(10-2.08)(0.7) =5.544 m/s

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