It has been suggested that rotating cylinders about 12.5 mi long and 4.88 mi in diameter be placed in space and used as colonies. What angular speed must such a cylinder have so that the centripetal acceleration at its surface equals the free-fall acceleration on Earth?
Answer in rad/s
To find the angular speed required for the rotating cylinder, we need to equate the centripetal acceleration at its surface to the free-fall acceleration on Earth.
The centripetal acceleration is given by the formula:
a = ω^2 * r
where:
a is the centripetal acceleration,
ω (omega) is the angular speed,
r is the radius.
In this case, the radius is half of the diameter, so r = 4.88 mi / 2 = 2.44 mi.
The free-fall acceleration on Earth is approximately 9.8 m/s^2. To convert miles to meters, we need to multiply this value by a conversion factor. 1 mile is equal to approximately 1609.34 meters.
Therefore, the free-fall acceleration on Earth is:
9.8 m/s^2 * (1 mi / 1609.34 m) = 0.006093 ft/s^2
To convert from feet per second squared to miles per hour squared, we need to use another conversion factor. 1 mile per hour is equal to approximately 1.467 ft/s.
So, the free-fall acceleration on Earth is:
0.006093 ft/s^2 * (1 mi / 1.467 ft/s)^2 = 0.00022227 mi/hr^2
Now, we can equate the centripetal acceleration to the free-fall acceleration and solve for ω:
0.00022227 mi/hr^2 = ω^2 * 2.44 mi
Rearranging the equation, we get:
ω^2 = 0.00022227 mi/hr^2 / 2.44 mi
ω^2 = 9.1021 * 10^-5 hr^-2
Taking the square root of both sides, we find:
ω ≈ ±0.00954 hr^-1
Since the angular speed cannot be negative in this context, the approximate angular speed required for the rotating cylinder is:
ω ≈ 0.00954 hr^-1
Now, we need to convert hours to seconds. 1 hour is equal to 3600 seconds.
So, the angular speed in rad/s is:
0.00954 hr^-1 * (3600 s / 1 hr) * (2π rad / 1 s) ≈ 21.1 rad/s
Therefore, the angular speed required for the rotating cylinder is approximately 21.1 rad/s.
First change 2.44 mi into meters (radius)
a = rw^2 so w = sqrt(a/r) = sqrt (9.8/radius)
You could also go "English" and change the miles to feet and use a = 32.