The 20-g centrifuge at NASA's Ames Research Center in Mountain View, California, is a horizontal, cylindrical tube 58.0 ft long and is represented in the figure below. Assume an astronaut in training sits in a seat at one end, facing the axis of rotation 29.0 ft away. Determine the rotation rate, in revolutions per second, required to give the astronaut a centripetal acceleration of 10.8g.
To determine the rotation rate required to give the astronaut a centripetal acceleration of 10.8g, we can use the following steps:
Step 1: Convert the acceleration from g to meters per second squared (m/s^2):
Since 1 g is equal to 9.8 m/s^2, the centripetal acceleration of 10.8g can be calculated as follows:
10.8g = 10.8 * 9.8 m/s^2 = 105.84 m/s^2.
Step 2: Find the distance from the axis of rotation to the astronaut:
In this case, the distance from the axis of rotation to the astronaut is given as 29.0 ft. To convert this to meters, we multiply by the conversion factor:
29.0 ft * 0.3048 m/ft = 8.8392 m.
Step 3: Use the formula for centripetal acceleration to find the rotation rate:
The formula for centripetal acceleration is given by: a = ω^2 * r, where a is the centripetal acceleration, ω is the rotation rate in radians per second, and r is the distance from the axis of rotation to the object.
Plugging in the known values:
105.84 m/s^2 = ω^2 * 8.8392 m.
Step 4: Solve for ω (rotation rate):
To isolate ω, we divide both sides of the equation by 8.8392 m and take the square root:
ω^2 = 105.84 m/s^2 / 8.8392 m.
ω^2 = 11.98 s^(-2).
ω = √(11.98 s^(-2)).
ω ≈ 3.46 s^(-1).
Therefore, the rotation rate required to give the astronaut a centripetal acceleration of 10.8g is approximately 3.46 revolutions per second.