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

think about the oscillation of the toy shown in the video. when the toy is at the highest point of its motion, its velocity is zero and its acceleration is negative.

c

explanation: