Question

An astronaut is rotated in a horizontal centrifuge at a radius of 3.90m.
a. What is the astronaut's speed if the centripetal acceleration has a magnitude of 7.50g?
b. How many revolutions per minute are required to produce this acceleration?
c. What is the period of the motion?

Answer 1

The astronaut's speed is approximately 16.93 m/s.

About 41.41 revolutions per minute are required to achieve a centripetal acceleration of 7.50g.

The period of the motion is approximately 1.45 seconds.

Let's solve the problem step by step:

1. Determine the speed of the astronaut:
   We know that : ac = v² / r
   Now, convert the acceleration to m/s² : ac = 7.50 * 9.81 m/s² = 73.575 m/s²
   Now solve for the speed, v:
   v² = ac * r
   v² = 73.575 m/s² * 3.90 m
   v² = 286.9425 m²/s²
   v = √286.9425 m²/s² ≈ 16.93 m/s
   The astronaut's speed is approximately 16.93 m/s.
2. Calculate the number of revolutions per minute (rpm):
   We know that one revolution covers a distance of the circumference, C = 2πr.
   C = 2π( 3.90 m) ≈ 24.50 m
   The number of revolutions per second (rps) is:
   rps = v / C = 16.93 m/s / 24.50 m ≈ 0.69 rps
   Convert rps to rpm:
   rpm = 0.69 rps * 60 s/min ≈ 41.41 rpm
   Approximately 41.41 revolutions per minute are required to produce the given acceleration.
3. Determine the period of the motion:
   The period (T) is the time for one complete revolution.
   We know:
   T = 1 / rps
   rps ≈ 0.69
   T ≈ 1 / 0.69 s ≈ 1.45 s
   The period of the motion is approximately 1.45 seconds.