Question

2. Artificial gravity is necessary for humans to stay healthy during extended stays in space. To provide artificial gravity, a space station is shaped like a cylinder with a diameter of 2200 m, and it rotates about its center at a constant rate. The astronauts live inside the cylinder at the outer edge, and feel a gravitational pull outwards. (a) How long should one revolution of the space station take if the acceleration of an astronaut on the outer edge of the cylinder is to have an acceleration of 9.8 m/s? Now suppose that an astronaut in the station holds a ball 2 m above the floor (the inner surface of the cylinder) and releases it when it is at rest relative to herself. From her point of view, it will drop to the floor similarly to what would happen on earth. Analyze this motion from the point of view of an inertial reference frame where the center of the space station is at rest and the astronaut is moving in uniform circular motion around it. In this reference frame the ball has an initial velocity when it is released. (b) What is the initial velocity of the ball in the inertial frame? (c) What is the acceleration of the ball after it is released in the inertial frame? (d) How long does it take for the ball to reach the floor in the inertial frame? (e) How long does it take a ball released from rest on the earth's surface to hit the ground? Are the times computed in parts (d) and (e) the same?

Answer 1

(A) a_c = w^2 r

9.8 = (2200/2) w^2

w = 0.0944rad/s

T = 2 pi / w = 66.6 sec ......Ans

(B) v = w r = (0.0944)(2200/2) = 103.8 m/s

(C) yf - yi = v0y t + ay t^2 /2 (initial velocity was in perpendicular direction)

0 - 2 = 0 - 9.8 t^2 /2

t = 0.64 sec

(D), yf - yi = v0y t + ay t^2 /2

0 - 2 = 0 - 9.8 t^2 /2

t = 0.64 sec

(e) same