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Rotating Space Stations. One problem for humans living in outer space is that they are apparently weightless. One way around this problem is to design a space station that spins about its center at a constant rate. This creates "artificial gravity" at the outside rim of the station. (a) If the diameter of the space station is 800 \(\mathrm{m}\) , how many revolutions per minute are needed for the "artificial gravity" acceleration to be 9.80 \(\mathrm{m} / \mathrm{s}^{2}\) (b) If the space station is a waiting area for travelers going to Mars, it might be desirable to simulate the acceleration due to gravity on the Martian surface \(\left(3.70 \mathrm{m} / \mathrm{s}^{2}\right) .\) How many revolutions per mimute are needed in this case?

Step by step solution

01

Understanding the Problem

We need to configure a rotating space station to simulate gravitational forces similar to that on Earth and Mars by calculating the revolutions per minute (RPM) required for specific acceleration levels. The space station has a diameter of 800 meters.

02

Calculate Radius of the Space Station

The diameter of the space station is 800 m. To find the radius, which we use in formulas for circular motion, divide the diameter by 2:\[ r = \frac{800}{2} = 400 \, \text{m} \]

03

Relate Centripetal Acceleration to RPM

Centripetal acceleration \( a_c \) is given by the formula:\[ a_c = \frac{v^2}{r} \]where \( v \) is the linear velocity and \( r \) is the radius. The linear velocity can be expressed in terms of the angular velocity \( \omega \) (in radians per second) as \( v = r \omega \). Thus,\[ a_c = r \omega^2 \]

04

Convert Angular Velocity to Revolutions Per Minute

The angular velocity \( \omega \) is in radians per second, but we need revolutions per minute for a complete understanding. The conversion uses the fact that one revolution is \( 2\pi \) radians, and there are 60 seconds in a minute:\[ \text{Revolutions per minute (RPM)} = \frac{\omega}{2\pi} \times 60 \]

05

Calculate RPM for Earth-like Gravity

For Earth-like artificial gravity \( a_c = 9.80 \, \text{m/s}^2 \):1. Solve for \( \omega \) using \( a_c = r \omega^2 \):\[ 9.80 = 400 \omega^2 \]\[ \omega^2 = \frac{9.80}{400} \]\[ \omega = \sqrt{\frac{9.80}{400}} \approx 0.15637 \, \text{rad/s} \]2. Convert \( \omega \) to\ RPM:\[ \text{RPM} = \frac{0.15637}{2\pi} \times 60 \approx 1.49 \]

06

Calculate RPM for Mars-like Gravity

For Mars-like artificial gravity \( a_c = 3.70 \, \text{m/s}^2 \):1. Solve for \( \omega \) using \( a_c = r \omega^2 \):\[ 3.70 = 400 \omega^2 \]\[ \omega^2 = \frac{3.70}{400} \]\[ \omega = \sqrt{\frac{3.70}{400}} \approx 0.09625 \, \text{rad/s} \]2. Convert \( \omega \) to\ RPM:\[ \text{RPM} = \frac{0.09625}{2\pi} \times 60 \approx 0.92 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rotational Motion

Rotational motion is a fundamental concept in physics that deals with objects revolving around a central point. In the case of our rotating space station, this means the station is spinning about its axis. As it spins, every point on the station follows a circular path.
This circular motion is characterized by two elements:

• Angular velocity (\( \omega \)) - measures how fast an object is rotating. It is measured in radians per second (rad/s).
• Radius of rotation - the distance from the center of rotation to a point on the edge of the circle. In our case, the radius is half the diameter of the station, which is 400 m.

Understanding these components helps in calculating other parameters like centripetal acceleration, which is crucial for achieving artificial gravity.

Artificial Gravity

Artificial gravity is a simulated gravitational force created by the rotation of space environments like space stations. In space, where natural gravitational pull is nearly absent, creating artificial gravity allows astronauts to function in a familiar environment.
To create artificial gravity:

• The space station spins at a calculated speed to generate an outward force. This force feels similar to gravity and is directed towards the station's walls, where people remain standing.
• The strength of this gravitational force can be controlled by adjusting the station's rotational speed. More speed means a stronger pull; less speed results in weaker force.

By simulating Earth-like gravity at 9.8 \( \text{m/s}^2 \) or Mars' gravity at 3.7 \( \text{m/s}^2 \), we set the spinning rate to ensure astronauts' well-being.

Centripetal Acceleration

Centripetal acceleration is the inward force needed to keep an object moving in a circular path. It determines how quickly the rotation needs to be adjusted to maintain that path. With the space station, this acceleration is crucial for creating the sensation of gravity.
Centripetal acceleration (\( a_c \)) depends on two factors:

• Angular velocity (\( \omega \)) - higher angular velocity means higher centripetal acceleration.
• Radius of the circular path - larger radius necessitates less acceleration for the same rotation speed.

The formula \[ a_c = r \omega^2 \] helps calculate the correct forces for specific gravitational simulations, such as Earth or Mars levels of gravity. Converting this result into revolutions per minute allows us to calibrate the station's rotation to achieve the desired artificial gravity.