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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?

Verified Answer :

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.

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