A projected space station consists of a circular tube that is set rotating about its center (like a tubular bicycle tire) as shown in figure below.
The circle formed by the tube has a diameter of about D = 1.28km. What must be the rotation speed (in revolutions per day) if an effect equal to gravity at the surface of the Earth (1g) is to be felt? Do not enter units.
To determine the rotation speed needed to create an effect equal to gravity (1g) at the surface of the Earth, we can use the concept of centripetal acceleration.
1. First, we need to find the radius of the circular tube. Since the diameter (D) is given as 1.28 km, we can calculate the radius (r) by dividing the diameter by 2:
r = D / 2 = 1.28 km / 2 = 0.64 km
2. Next, we need to convert the radius from kilometers to meters, as the standard unit for acceleration is meters:
r = 0.64 km * 1000 = 640 meters
3. The formula for centripetal acceleration is a = v^2 / r, where "a" is the acceleration, "v" is the velocity, and "r" is the radius.
4. Since we want the effect of gravity (1g), we know that the centripetal acceleration must be equal to the acceleration due to gravity on Earth, which is approximately 9.8 m/s^2.
5. By substituting the values into the formula, we get:
9.8 = v^2 / 640
6. To solve for v, we need to rearrange the equation:
v^2 = 9.8 * 640
v^2 = 6272
7. Taking the square root of both sides:
v ≈ √6272
v ≈ 79.13 m/s
8. Finally, to convert the velocity to revolutions per day, we need to relate the distance traveled in one revolution to the circumference of the circle. The circumference (C) can be calculated using the formula C = 2πr:
C = 2 * π * 640
9. To find the number of revolutions per day, we divide the distance traveled in one day by the circumference of the circle:
Revolutions per day ≈ 288,000 meters (distance in one day) / C
By following these steps, you can calculate the rotation speed (in revolutions per day) required for the projected space station.