Question

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/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 (3.70 m/s\(^2\)). How many revolutions per minute are needed in this case?

Short answer

(a) 1.50 rev/min, (b) 0.92 rev/min

Step-by-step solution

Understanding the problem

We need to calculate how many revolutions per minute are required for a rotating space station to produce specific accelerations: Earth's gravity (9.80 m/s²) and Martian gravity (3.70 m/s²).

Determine the radius of the space station

Since the diameter of the space station is 800 m, the radius is half of the diameter. \[ \text{Radius} = \frac{800}{2} = 400\,m \]

Relate centripetal acceleration to angular velocity

The formula for centripetal acceleration, \( a \), in terms of angular velocity, \( \omega \), and radius, \( r \), is: \[ a = \omega^2 \times r \] We will solve for \( \omega \).

Solve for angular velocity for Earth's gravity

Substitute \( a = 9.80 \text{ m/s}^2 \) and \( r = 400 \text{ m} \) into the formula to find \( \omega \):\[ 9.80 = \omega^2 \times 400 \]\[ \omega^2 = \frac{9.80}{400} \]\[ \omega \approx 0.157 \text{ rad/s} \]

Convert angular velocity to revolutions per minute for Earth's gravity

Convert \( \omega \) from radians per second to revolutions per minute. 1 revolution = \( 2\pi \) radians, and there are 60 seconds in a minute. \[ \text{revolutions/minute} = \frac{0.157 \times 60}{2\pi} \approx 1.50 \text{ rev/min} \]

Solve for angular velocity for Martian gravity

Substitute \( a = 3.70 \text{ m/s}^2 \) and \( r = 400 \text{ m} \) into the formula to find \( \omega \).\[ 3.70 = \omega^2 \times 400 \]\[ \omega^2 = \frac{3.70}{400} \]\[ \omega \approx 0.096 \text{ rad/s} \]

Convert angular velocity to revolutions per minute for Martian gravity

Convert \( \omega \) from radians per second to revolutions per minute.\[ \text{revolutions/minute} = \frac{0.096 \times 60}{2\pi} \approx 0.92 \text{ rev/min} \]

Key concepts

Centripetal Acceleration

Centripetal acceleration is a critical concept when discussing artificial gravity in space stations. It refers to the acceleration experienced by an object moving along a circular path, directed towards the center of the circle. This acceleration changes the object's direction, keeping it on the circular path without altering its speed.
In the context of a rotating space station, centripetal acceleration is what simulates gravity on Earth or Mars for the inhabitants. The sensation of gravity produced is due to the force required to keep inhabitants moving in a circle within the station. This force is vital to maintain normal activities and well-being, mimicking gravity even in space.
To calculate centripetal acceleration, you use the formula: \( a = \omega^2 \times r \), where \( \omega \) is angular velocity, and \( r \) is the radius of the space station. This formula helps determine the rate at which the station should spin to achieve desired levels of gravity-like effects.

Rotational Motion

Rotational motion is integral to creating artificial gravity in space stations. It involves the motion of rotating about an axis. This type of motion is what allows the space station to simulate gravity for its occupants.
In a rotating space station, every point along the outer rim travels in a circular path. The speed and direction of this motion dictate the "gravity" experienced by the occupants. Understanding rotational motion helps in designing stations with efficient force distribution across all points on its rim.
Key elements of rotational motion include the radius of rotation, angular velocity, and centripetal force. Each element influences how effectively a space station maintains a stable environment for its inhabitants, providing predictable and consistent artificial gravity.

Angular Velocity

Angular velocity is a term used to describe how fast an object rotates or spins around a particular point or axis. It is crucial for determining the rate at which a space station must rotate to provide artificial gravity.
The angular velocity \( \omega \) is measured in radians per second. In space station design, this measure helps ensure that the rotation is neither too slow nor too fast to simulate the desired gravity level accurately.
To convert angular velocity from radians per second to revolutions per minute, the process involves converting radians (with \( 2\pi \) radians equaling one full revolution) and accounting for the time conversion from seconds to minutes. For example, an angular velocity of 0.157 rad/s translates to approximately 1.5 revolutions per minute, which corresponds to Earth's gravity conditions.

Space Station Design

Designing a space station with artificial gravity capability involves thoughtful consideration of structural elements. These designs utilize rotational dynamics to simulate gravitational effects through controlled spinning.
Key design factors include the station's diameter and rotational speed. For instance, a larger diameter allows a slower rotation rate to achieve the same artificial gravity level. This is due to the larger circumference providing a greater linear velocity at the rim for the same angular velocity.

• Choice of materials: Must withstand the stresses of constant rotation.
• Energy efficiency: Devices to maintain rotation use energy; efficient designs minimize this.
• Safety: Must ensure that the rotational mechanics pose no hazard to inhabitants.

The goal in space station design is to mimic Earth-like conditions as closely as possible, promoting comfort and health for long-term space dwellers. Rotation introduces challenges that must be addressed creatively for a successful design.