In the design of a space station to operate outside the earth’s gravitational fi eld, it is desired to give the structure a rotational speed n which will simulate the effect of the earth’s gravity for members of the crew. if the centers of the crew’s quarters are to be located 12 m from the axis of rotation, calculate the necessary rotational speed n of the space station in revolutions per minute.

For an object in circular motion, we have this formula for centripetal acceleration:

a = v²/r

a is the centripetal acceleration, v is the rotational speed, and r is the radius of the circular path.

Given values:

a = 9.81m/s² (we want the centripetal acceleration to be the same as the acceleration due to gravity on earth)

r = 12m

Plug in the values and solve for N:

9.81 = v²/12

v² = 117.72

v = ±10.8m/s

Reject the negative value.

v = 10.85m/s

Now we want to obtain the angular velocity. It is related to the rotational speed by:

v = rω

v is the rotational speed, r is the radius of rotation, and ω is the angular velocity.

10.85 = 12ω

ω = = 0.9041rad/s

We'll use the conversion rate 1 rad/s = 9.55rpm:

0.9041(9.55)

= 8.63rpm