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 as shown in the figure below. The cabins are set spinning around the hub axis, which is connected to the rest of the spacecraft to generate artificial gravity.

(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. (Use the following as necessary: m, v, and r.)

Fnet = n = A space habitat for a long space voyage consists of two cabins each connected by a cable to a central hub as shown in the figure below. The cabins are set spinning around the hub axis, which is connected to the rest of the spacecraft to generate artificial gravity. What forces are acting on an astronaut in one of the cabins? 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. (Use the following as necessary: m, v, and r.) What would n have to equal if the 51.8-kg astronaut is to experience half his normal Earth weight? n = N Calculate the necessary tangential speed of the habitat from Newton's second law. m/s Calculate the angular speed from the tangential speed. rad/s Calculate the period of rotation from the angular speed. s If the astronaut stands up, will his head be moving faster, slower, or at the same speed as his feet? Why? Calculate the tangential speed at the top of his head if he is 1.59 m tall. m/s

Explanation / Answer

(a) What forces are acting on an astronaut in one of thecabins? Since there must be a force that keeps the astronaut in therotating circle, we know that the net force is towards the centerof the circle. Since there is no gravity, the only force must bethe centripetal force (Normal force) towards the center of thehub.

(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 hubr, and the normal force n. Newton's second law: mv^2/r normal force: mv^2/r

(c) What would n have to equal if the 51.8-kg astronaut is to experience half his normalEarth weight? Normal force=ma Normal force=(51.8)(9.8)=507.64 Half Normal Force=507.64/2=253.82

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

F=mv^2/r

253.82=(51.8)(v^2)/10

2538.2=(51.8)(v^2)

49=v^2

v=7m/s

(e) Calculate the angular speed from the tangential speed. w=v/r w=7/10 radian/sec

(f) Calculate the period of rotation from the angular speed. Period=2pi/.7 =9.976

(g) If the astronaut stands up, will his head be moving faster,slower, or at the same speed as his feet? Why? The astronaut's head will be moving faster than his feet becausethe radius from his head to the center of the hub is shorter thanthat from his feet to the hub.

(i) Calculate the tangential speed at the top of his head if he is1.59 m tall.

v=(r-h)w

v=(10-1.59)(.7)

v=5.887

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