(II) A proposed space station consists of a circular tube that will rotate about its center (like a tubular bicycle tire), Fig. 5–47. The circle formed by the tube has a diameter of about 1.1 km. What must be the rotation speed (revolutions per day) if an effect nearly equal to gravity at the surface of the Earth, 0.90 g, is to be felt by astronauts walking inside? Which part of the tube do they walk on?
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Verified step by step guidance

1. Identify the radius of the circular tube. Since the diameter is given as 1.1 km, the radius (r) will be half of the diameter, i.e., 0.55 km.
2. Understand that the astronauts will experience a centrifugal force that acts as artificial gravity when the station rotates. They will walk on the inside surface of the tube, facing towards the center.
3. Use the formula for centripetal acceleration, which is a = $\frac{v^2}{r}$, where v is the linear speed and r is the radius. The desired artificial gravity is 0.90 g, where g is the acceleration due to gravity (approximately 9.8 m/s^2). Set up the equation $0.90\times 9.8=\frac{v^2}{0.55\times 1000}$ to find v.
4. Convert the linear speed (v) into angular velocity ($\omega$) using the relationship $\omega =\frac{v}{r}$.
5. Finally, convert the angular velocity from radians per second to revolutions per day to find the required rotation speed of the space station.

Key Concepts

Centrifugal Force

Centrifugal force is an apparent force that acts outward on a mass moving in a circular path, arising from the inertia of the mass. In the context of the rotating space station, this force simulates gravity for astronauts inside the tube. The required speed of rotation can be calculated to ensure that the centrifugal force equals the gravitational force experienced on Earth.

Angular Velocity

Angular velocity is a measure of how quickly an object rotates around a central point, expressed in radians per second or revolutions per unit time. For the space station, determining the angular velocity is crucial to calculate the necessary rotation speed to create an artificial gravity effect equivalent to 0.90 g. This involves understanding the relationship between linear speed and angular velocity.

Artificial Gravity

Artificial gravity refers to the simulation of gravitational effects in a non-gravitational environment, such as space. In the proposed space station, astronauts experience artificial gravity through the centrifugal force generated by the station's rotation. The design must ensure that the rotation speed is sufficient to create a force that mimics Earth's gravity, allowing astronauts to walk comfortably inside the tube.