The 20-g centrifuge at NASA's Ames Research Center in Mountain View, California, is a horizontal, cylindrical tube 58 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 19.0g.
please explain steps/thinking thank you!
To determine the rotation rate required to give the astronaut a centripetal acceleration of 19.0g, we can use the following formula:
centripetal acceleration (a) = (v^2) / r
Where:
a = centripetal acceleration
v = tangential velocity
r = radius of rotation
In this case, the centripetal acceleration is given as 19.0g, and we need to find the rotation rate, which is equivalent to the tangential velocity (v) in this scenario.
To convert the acceleration from g to meters per second squared (m/s^2), we use the conversion factor: 1g = 9.8 m/s^2.
Thus, the centripetal acceleration becomes:
a = 19.0g * 9.8 m/s^2
Now, let's calculate the tangential velocity (v):
v = √(a * r)
where r is the distance between the astronaut and the center of rotation, given as 29.0 ft.
First, we convert the distance from feet to meters:
r = 29.0 ft * 0.3048 m/ft
Now, we can calculate the tangential velocity (v):
v = √(a * r)
Finally, to find the rotation rate in revolutions per second, we divide the tangential velocity by the circumference of the cylindrical tube, which is given as 58 ft.
Let's calculate all the values step by step:
Step 1: Conversion of acceleration
a = 19.0g * 9.8 m/s^2
Step 2: Conversion of distance
r = 29.0 ft * 0.3048 m/ft
Step 3: Calculation of tangential velocity
v = √(a * r)
Step 4: Calculation of rotation rate
rotation rate = v / (2 * π * r)
Now, let's substitute the values and calculate the rotation rate.
Centripetal acceleration is
a = ω ²•r
where ω is the angular velocity (in radians per second), and r is the radius r=29 ft
ω=sqrt(a/r)= …. (in rad/s)
f(rev/s)= ω (rad/s)/2π