In order to produce an artificial gravity in the space craft, the laboratory of space craft is rotated with suitable frequency about its own axis. The rotation is so maintained that the astronaut do not feel weightlessness. The frequency of rotation depends on the length of laboratory of space craft.
Consider a space craft whose laboratory is 'L' meter long consisting of two chambers connected by a tunnel. Let us see how many revloutions per second must the space craft make in order to supply artificial gravity for the astronauts.
Let 'T' be the time for one revolution and 'f' be the frequency of rotation.
R = L / 2 --------(i)
When the laboratory revolves, a centripetal force is experienced by the astronauts.
Fc = mv2/r --------(ii)
According to Newton's second law of motion
Fc = mac --------(iii)
Comparing equations (ii) and (iii)
mac = mv2/r
ac = v2/r
Where ac is the centripetal acceleration
Since radius of laboratory is R , therefore,
ac = v2/R --------(iv)
Now we will determine the linear speed of the laboratory.
In one rotation of the laboratory
Distance = 2pR
time = T
velocity = ?
Using the relation s = vt
2pR = vT
v = 2pR/T
Putting the value of v in equation (iv), we get,
ac = (2pR/T)2/R
ac = (4p2R2/T2)/R
ac = (4p2R/T2)
ac = 4p2R x 1/T2
But 1/T = frequency (f)
ac = 4p2R x f2
f2 = ac/4p2R
For natural gravity acceleration must be equal to 9.8m/s2 i.e. ac = g , thus
This expression indicates that the frequency of rotation depends on the length of the laboratory of space craft. Larger is the laboratory, smaller should be the number of rotation per second to obtain the natural gravity effect.