Question

One way to provide artificial gravity (i.e., a feeling of weight) on long space voyages is to separate a spacecraft into two parts at the ends of a long cable, and set them rotating about their center of mass. A craft has been separated into two parts with a mass of 66900 kg each, at the ends of a cable with their centers of mass 139 m apart, rotating around the center point of the cable with a period of 233.5 seconds.

The two pieces of the spacecraft are returned to their initial configuration with a period of 233.5 seconds. A radiation leak is detected in one of the engines. An escape pod from each section of the spacecraft are ejected as indicated in the figure. Notice that the direction of rotation for the two pieces of the spacecraft is also indicated in the figure.

The escape pods each have a mass of 10700 kg and are ejected with a speed of 130.95 m/s along the directions indicated in the figure with ? = 31 degrees. What is the new period of the remaining spacecraft after the two escape pods are ejected?

Answer 1

given space craft

mass of each part , m = 66900 kg

distance between center of masses, d = 139 m

Time period, T = 233.5 s

now, mass of each escape pod, m' = 10,700 kg

direction, theta = 31 deg

escape speed, v = 130.95 m/s

a. if the pods seperate in anti parallel direction

then final angular speed of the space station = w'

T' = 2*pi/w'

hence fromconservation of angular momentum

2m(d/2)^2*w = 2(m - m')(d/2)^2*w' + 2*m'*cos(theta)*v

md^2*w = (m - m')d^2*w' + 4*m'*cos(theta)*v

w = 2*pi*/T

md^2*2*pi/T = (m - m')d^2*2*pi/T' + 4m'*cos(theta)v

hence

T' = 227.58928815 s

b. if the pods seperate in parallel direction

then period of rotation of the sapce station dies not change