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 m, how many revolutions per minute are needed for the 'artificial gravity' acceleration to be 9.80 m/s2?

Verified step by step guidance

1. Identify the given values: diameter of the space station (D) is 800 m, and the desired artificial gravity acceleration (a) is 9.80 m/s^2.
2. Calculate the radius (R) of the space station by dividing the diameter by 2. This gives R = D / 2.
3. Use the formula for centripetal acceleration, which is a = \( \frac{v^2}{R} \), where v is the tangential speed at the rim. Rearrange this formula to solve for v: v = \( \sqrt{a \times R} \).
4. Convert the tangential speed (v) to angular velocity (\( \omega \)) using the relationship \( \omega = \frac{v}{R} \).
5. Finally, convert the angular velocity (\( \omega \)) from radians per second to revolutions per minute (RPM). Use the conversion factor that 1 revolution is \( 2\pi \) radians and 1 minute is 60 seconds.

Key Concepts

Centripetal Acceleration

Centripetal acceleration is the acceleration directed towards the center of a circular path that keeps an object moving in that path. It is calculated using the formula a = v²/r, where 'a' is the centripetal acceleration, 'v' is the tangential velocity, and 'r' is the radius of the circular path. In the context of the space station, this acceleration mimics the effect of gravity for occupants.

Angular Velocity

Angular velocity is a measure of how quickly an object rotates around a central point, typically expressed in radians per second or revolutions per minute (RPM). It is related to linear velocity and radius by the equation v = ωr, where 'ω' is the angular velocity. Understanding angular velocity is crucial for determining how fast the space station must spin to achieve the desired artificial gravity.

Artificial Gravity

Artificial gravity refers to the simulation of gravitational effects in a non-gravitational environment, such as space. It can be achieved through centripetal force generated by rotating structures, like a spinning space station. The goal is to create a force that mimics Earth's gravity, allowing astronauts to experience similar conditions to those on Earth, which is vital for their health and well-being during long missions.