Question

A donut-shaped space station (outer radius \(R\) ) arranges for artificial gravity by spinning on the axis of the donut with angular velocity \(\omega .\) Sketch the forces on, and accelerations of, an astronaut standing in the station (a) as seen from an inertial frame outside the station and (b) as seen in the astronaut's personal rest frame (which has a centripetal acceleration \(A=\omega^{2} R\) as seen in the inertial frame). What angular velocity is needed if \(R=40\) meters and the apparent gravity is to equal the usual value of about \(10 \mathrm{m} / \mathrm{s}^{2} ?\) (c) What is the percentage difference between the perceived \(g\) at a six-foot astronaut's feet \((R=40 \mathrm{m})\) and at his head \((R=38 \mathrm{m}) ?\)

Short answer

(a) The centripetal force in the inertial frame is inward. (b) The astronaut feels outward force. (c) The required angular velocity is 0.5 rad/s and the perceived g difference is 5%.

Step-by-step solution

Understanding the Forces from an Inertial Frame

When viewed from an inertial frame outside the station, the astronaut is undergoing circular motion along the inner surface of the donut-shaped station. This means there is a centripetal force directed towards the center of the circular path, which accounts for the centripetal acceleration. This force is provided by the normal force from the station's surface.

Analyzing from the Astronaut's Rest Frame

In the astronaut's personal rest frame, the astronaut feels a force pushing outward—perceived as artificial gravity. This is the same magnitude as the centripetal force observed in the inertial frame, but here it appears as a force due to acceleration 'A' which pushes the astronaut against the floor of the station.

Calculating Angular Velocity for Artificial Gravity

For the apparent gravity to equal Earth's gravity (approximately 10 m/s²), the angular velocity \( \omega \) must cause a centripetal acceleration equal to this value. Using the formula for centripetal acceleration \( A = \omega^2 R \), we set \( A = 10 \mathrm{m/s^2} \). This gives \( \omega^2 = \frac{10}{40} = 0.25 \). Solving for \( \omega \) yields \( \omega = \sqrt{0.25} = 0.5 \, \mathrm{rad/s} \).

Calculating the Perceived Gravity Difference

The difference in perceived gravity results from the difference in radial positions between the astronaut's feet (\(R = 40\) m) and head (\(R = 38\) m). For the feet: \( g_f = \omega^2 \times 40 = 0.25 \times 40 = 10 \mathrm{m/s^2} \). For the head: \( g_h = \omega^2 \times 38 = 0.25 \times 38 = 9.5 \mathrm{m/s^2} \). The percentage difference is \( \frac{10 - 9.5}{10} \times 100\% = 5\% \).

Key concepts

Centripetal Force

Centripetal force keeps an object moving in a circular path by pushing it towards the center. In the space station example, an astronaut experiences this force as the station spins. The structure of the station keeps the astronaut moving in a circle, and this force prevents them from flying off. It acts perpendicular to the direction of the astronaut's velocity. Without it, the astronaut would move in a straight line instead of a circle.

Inside the station, the centripetal force is provided by the normal force, which is the contact force from the station's floor pressing against the astronaut. This setup creates the sensation of gravity pushing them against the floor. So even though the force acts inward, the astronaut perceives it as a downward force similar to gravity on Earth.

Angular Velocity

Angular velocity, denoted by \( \omega \), describes how fast something spins around a central point. In our spinning space station, \( \omega \) determines how quickly the astronaut completes a circular path. It's expressed in radians per second.

The centripetal acceleration experienced depends on both \( \omega \) and the radius \( R \) of the station. This is captured in the formula \( A = \omega^2 R \). For artificial gravity to mimic Earth's gravity, the station's \( \omega \) must be carefully adjusted so that the rotation creates a centripetal acceleration of approximately 10 m/s². This is accomplished by having \( \omega = 0.5 \) rad/s when \( R = 40 \) meters.

Inertial Frame

An inertial frame is a perspective from which objects viewed tend not to have acceleration unless acted upon by an external force. It's like watching a merry-go-round while standing still on the ground. From this perspective, all forces and their effects can be observed without experiencing them yourself.

When analyzing the space station from an inertial frame outside, you observe the astronaut being pulled towards the center. However, from the astronaut's perspective—non-inertial because they experience acceleration—the forces are perceived differently. They feel a force pressing them against the floor, an illusion of gravity created by the station's spin.

• This relates to Einstein's equivalence principle, suggesting that acceleration can mimic gravitational force.
• Within an inertial frame, it’s easier to calculate forces using classical physics.
• In the space station, moving at the right angular velocity creates an accurate simulation of gravity inside.