At its Ames Research Center, NASA uses its large '20-G' centrifuge to test the effects of very large accelerations ('hypergravity') on test pilots and astronauts. In this device, an arm 8.84 m long rotates about one end in a horizontal plane, and an astronaut is strapped in at the other end. Suppose that he is aligned along the centrifuge's arm with his head at the outermost end. The maximum sustained acceleration to which humans are subjected in this device is typically 12.5g. (c) How fast in rpm (rev/min) is the arm turning to produce the maximum sustained acceleration?

Verified step by step guidance

1. Identify the given values: the length of the centrifuge arm, L = 8.84 m, and the maximum sustained acceleration, a = 12.5g, where g is the acceleration due to gravity (approximately 9.8 m/s^2).
2. Calculate the maximum sustained acceleration in m/s^2 by multiplying the given acceleration in g's by the acceleration due to gravity: a = 12.5 \times 9.8 \, \text{m/s}^2.
3. Use the formula for centripetal acceleration, a = \frac{v^2}{r}, where v is the linear velocity and r is the radius (length of the arm). Rearrange the formula to solve for v: v = \sqrt{a \times r}.
4. Convert the linear velocity (v) to angular velocity (\omega) using the relationship \omega = \frac{v}{r}.
5. Convert the angular velocity from radians per second to revolutions per minute (rpm) using the conversion factor: 1 \, \text{rad/s} = \frac{60}{2\pi} \, \text{rpm}.

Key Concepts

Centripetal Acceleration

Centripetal acceleration is the acceleration directed towards the center of a circular path that keeps an object moving in that path. It is calculated using the formula a = v²/r, where 'a' is the centripetal acceleration, 'v' is the tangential speed, and 'r' is the radius of the circular path. In the context of the centrifuge, this acceleration is crucial for understanding how the rotation affects the astronaut.

Gravitational Force and g-Forces

The term 'g-force' refers to the force of gravity on an object, expressed in multiples of Earth's gravitational acceleration (approximately 9.81 m/s²). When subjected to acceleration, such as in a centrifuge, individuals experience forces greater than 1g, which can affect their physical condition. In this scenario, the maximum sustained acceleration of 12.5g indicates that the astronaut experiences 12.5 times the force of gravity due to the centrifuge's rotation.

Angular Velocity and RPM

Angular velocity is a measure of how quickly an object rotates around a central point, typically expressed in radians per second or revolutions per minute (RPM). To find the RPM of the centrifuge arm, one can relate the linear speed at the end of the arm to the angular velocity using the formula v = ωr, where 'ω' is the angular velocity and 'r' is the radius. This relationship is essential for calculating the speed required to achieve the specified acceleration.