Question

A biologist is studying growth in space. He wants to simulate Earth's gravitational field, so he positions the plants on a rotating platform in the spaceship. The distance of each plant from the central axis of rotation is \(r=0.20 \mathrm{m} .\) What angular speed is required?

Short answer

The required angular speed is approximately 7 rad/s.

Step-by-step solution

Understand the Problem Context

The biologist wants to simulate Earth's gravity, so the net centripetal force on the plant due to the platform's rotation should equal the gravitational force on Earth. We need to calculate the angular speed, \( \omega \), that accomplishes this.

Recall Centripetal Acceleration Formula

The formula for centripetal acceleration \( a_c \) is \( a_c = \omega^2 \times r \). Here, \( \omega \) is the angular speed and \( r \) is the radius of rotation, which is 0.20 m in this problem.

Set Centripetal Acceleration Equal to Gravitational Acceleration

In order for the plant to experience an acceleration equivalent to gravity on Earth, we set the centripetal acceleration equal to the gravitational acceleration \( g = 9.81 \text{ m/s}^2 \). This gives the equation \( \omega^2 \times r = g \).

Solve for Angular Speed (\( \omega \))

Rearrange the equation \( \omega^2 \times r = g \) to solve for \( \omega \). This becomes \( \omega = \sqrt{\frac{g}{r}} \).

Calculate Angular Speed

Substitute the known values into the equation. Use \( g = 9.81 \text{ m/s}^2 \) and \( r = 0.20 \text{ m} \). Therefore, \( \omega = \sqrt{\frac{9.81}{0.20}} = \sqrt{49.05} \approx 7 \text{ rad/s} \).

Key concepts

Centripetal Acceleration

When an object rotates in a circle, it needs a force to keep it moving along the curved path. This force is directed towards the center and is known as the centripetal force. The acceleration that results from this force is called centripetal acceleration. It measures how quickly the direction of the object's velocity is changing.

Centripetal acceleration can be calculated using the formula:

• \( a_c = \omega^2 \times r \)

Here, \( a_c \) represents centripetal acceleration, \( \omega \) is the angular speed, and \( r \) is the distance from the center of rotation.

For the plant on the spacecraft's rotating platform, the centripetal acceleration must match Earth's gravitational acceleration (\( 9.81 \text{ m/s}^2 \)) to simulate the gravity a plant would feel on Earth. By adjusting the angular speed, we can ensure the centripetal force substitutes Earth's gravitational pull.

Gravitational Field Simulation

Simulating Earth's gravitational field in space is crucial for conducting experiments on living organisms, like plants, that are accustomed to Earth's gravity. On a spacecraft, normal gravity is absent, which poses challenges for biological and physical processes that depend on gravity.

By using a rotating platform, we can mimic gravitational effects through centripetal force. By adjusting the angular speed of the platform in relation to the distance from the axis of rotation, we can create a centripetal acceleration equal to Earth's gravitational field. This is known as generating artificial gravity and allows researchers to study processes under familiar force conditions, even in zero-gravity environments like space.

Rotational Dynamics

Rotational dynamics deals with the study of objects rotating around a central point or axis. This concept is rooted in Newton's laws of motion but is adapted for rotating systems.

Key variables in rotational dynamics include:

• Angular speed (\( \omega \)), which measures how quickly an object rotates.
• Radius (\( r \)), which is the distance from the rotation axis.
• Centripetal force, keeping the object moving in a circular path.

In our context, by manipulating the angular speed and distance from the axis, we can control the centripetal acceleration and simulate various gravitational conditions. With the right rotational dynamics, it is possible to create an environment in space that feels like Earth, which is essential for many space-based experiments.

Space Conditions

Space offers a unique environment vastly different from Earth. These conditions include a lack of atmosphere, microgravity, and the absence of a natural gravitational pull. Such an environment impacts how objects move and how biological processes occur.

To adapt, spacecraft often simulate gravity using rotational devices. In our exercise, the platform's rotation creates centripetal acceleration, offering an effective way to simulate gravity for biological experiments.

Understanding space conditions is crucial for planning experiments that rely on Earth-like interactions, ensuring safety, and obtaining valid data. As we explore space further, simulating home-like conditions through physics becomes more vital for sustaining long-term missions and understanding living systems beyond our planet.