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. 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. Step 1: Understand the concept of artificial gravity. Artificial gravity is created by the centripetal acceleration experienced by objects on the rim of a rotating space station. The centripetal acceleration is given by the formula: $a=r{\omega }^2$, where $a$ is the centripetal acceleration, $r$ is the radius of the space station, and $\omega$ is the angular velocity.
2. Step 2: Calculate the radius of the space station. The diameter is given as 800 m, so the radius is half of the diameter: $r=\frac{800}{2}=400\text{m}$.
3. Step 3: Rearrange the centripetal acceleration formula to solve for angular velocity $\omega$. The formula becomes: $\omega =\sqrt{\frac{a}{r}}$. Substitute the values for $a$ (9.80 m/s²) and $r$ (400 m) into the formula.
4. Step 4: Convert angular velocity $\omega$ to revolutions per minute (rpm). Angular velocity is typically measured in radians per second, so use the conversion factor: $\frac{1}{\mathrm{2\pi }}$ revolutions per radian and multiply by 60 seconds per minute. The formula for rpm is: $\mathrm{rpm}=\frac{60}{\mathrm{2\pi }}\omega$.
5. Step 5: Substitute the calculated angular velocity $\omega$ into the rpm formula to find the number of revolutions per minute required for the artificial gravity acceleration to be 9.80 m/s².

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 '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 effects 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 through the equation v = ωr, where 'ω' is the angular velocity and 'r' is the radius. 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. This is achieved through centripetal force generated by rotating structures, like a spinning space station. By creating a force that acts outward from the center of rotation, occupants experience a sensation similar to gravity, which is essential for their physical well-being during long-term space missions.