Problem 32 - College Physics

Question

A rotating cylinder about \(16 \mathrm{~km}\) long and \(7.0 \mathrm{~km}\) in diameter is designed to be used as a space colony. With what angular speed must it rotate so that the residents on it will experience the same acceleration due to gravity as on Earth?

Step-by-Step Solution

Answer

The angular speed must be approximately \(0.053 \text{ rad/s}\).

Step 1: Understand the Problem

We want the residents inside a rotating space cylinder to experience the same gravitational force as on Earth. This means we need to generate an acceleration equal to the acceleration due to gravity on Earth's surface, which is approximately \(9.81 \text{ m/s}^2\).

Step 2: Relate Circular Motion to Gravity

The artificial gravity is generated by the centripetal acceleration of the cylinder's rotation. The formula for centripetal acceleration \(a_c\) is given by \(a_c = \omega^2 r\), where \(\omega\) is the angular speed we want to find, and \(r\) is the radius of the cylinder (half of the diameter).

Step 3: Calculate the Radius of the Cylinder

The diameter of the cylinder is \(7.0 \text{ km}\). Convert this to meters and find the radius. \(7.0 \text{ km} = 7000 \text{ m}\), so the radius \(r = \frac{7000 \text{ m}}{2} = 3500 \text{ m}\).

Step 4: Solve for Angular Speed

Set the centripetal acceleration equal to the gravitational acceleration on Earth: \(\omega^2 r = 9.81\). Rearrange to solve for \(\omega\): \(\omega = \sqrt{\frac{9.81}{r}}\).

Step 5: Calculate Angular Speed

Substitute \(r = 3500 \text{ m}\) into the formula: \(\omega = \sqrt{\frac{9.81}{3500}}\). Calculate to find \(\omega \approx 0.053 \text{ rad/s}\).

Step 6: Confirm the Solution

Verify that the calculated angular speed, \(\omega \approx 0.053 \text{ rad/s}\), when used in \(\omega^2 r\) equals the gravitational acceleration \(9.81 \text{ m/s}^2\). This confirms the solution is correct.

Key Concepts

Centripetal Acceleration

Centripetal acceleration is the acceleration that keeps an object moving in a circular path. It is directed towards the center of the circle. In the context of artificial gravity, it is this inward acceleration that creates a sensation similar to gravity. When a rotating space colony, like the designed cylinder, spins, the inhabitants feel as though gravity is pulling them towards the outer edge of the cylinder. This is because the floor of the cylinder exerts a force that acts as the centripetal force.
To understand how this works, consider the formula for centripetal acceleration:

• \[ a_c = \omega^2 r \]
• Here, \( a_c \) is the centripetal acceleration, \( \omega \) is the angular speed, and \( r \) is the radius of the circular path.

Generating enough centripetal acceleration to mimic Earth's gravity requires the right angular speed for the radius of the colony. This relationship allows us to design space habitats where life feels just like it does on Earth.

Angular Speed

Angular speed is the rate at which an object rotates around a circular path. It is an essential factor in achieving artificial gravity in a rotating space colony. The unit for angular speed is typically radians per second (rad/s).
For the space cylinder in our problem, the objective is to find the angular speed that produces a centripetal acceleration equal to Earth's gravity (\(9.81 \, ext{m/s}^2\)). The formula to calculate angular speed when given centripetal acceleration is:

• \[ \omega = \sqrt{\frac{a_c}{r}} \]

Here, \( a_c \) is the desired centripetal acceleration, and \( r \) is the radius of the cylinder. By substituting the known values of \( a_c \) and \( r \) derived from the diameter, we compute the ideal angular speed that renders the experience of regular Earth gravity inside the space habitat. This is crucial for space colonization as it ensures comfort and familiar living conditions for residents.

Circular Motion

Circular motion is motion along a circular path, and it plays a pivotal role in creating artificial gravity within space colonies. When an object moves in a circle, its direction is constantly changing, meaning it always experiences acceleration known as centripetal acceleration. This concept is indispensable in the design of rotating habitats.
The essential aspects of circular motion in this context include:

• An object moves with constant speed along the circle.
• The direction of the motion changes at every point on the path.
• Centripetal force acts perpendicular to the velocity at any point, ensuring the change in direction.

In a rotating cylinder space habitat, circular motion ensures every part of the cylinder moves to simulate gravity uniformly across its radius. This feat is accomplished by carefully calibrating the rotation speed to maintain a consistent centripetal force throughout the living space.

Space Colonization

Space colonization involves creating self-sustaining human habitats beyond Earth. As the plausibility of living on other planets grows, one of the big challenges is creating environments that mimic Earth's conditions, especially gravity, which is crucial for human health and well-being. Designing rotating space habitats is one solution to this problem.
By utilizing concepts like centripetal acceleration and angular speed, scientists aim to produce artificial gravity. This artificial gravity allows humans to live in comfort, conducting daily activities without the debilitating effects of microgravity.

• Space colonies may take various forms, but rotating structures are among the most viable options for generating a gravity-like environment.
• Besides physical comfort, artificial gravity is crucial for conserving muscle and bone density, vital for long-term human health.
• With advancements in technology, these habitats could house vast populations, supporting agriculture, industry, and research in space.

Overall, space colonization holds the potential to extend human civilization beyond Earth, ensuring survival and fostering growth as space exploration continues to advance.