Question

An astronaut is rotated in a centrifuge of radius \(5.2 \mathrm{~m} .(a)\) What is the speed if the acceleration is \(6.8 g ?(b)\) How many revolutions per minute are required to produce this acceleration?

Short answer

The speed of rotation is approximately 24.69 m/s, and the centrifuge needs to make approximately 45.19 revolutions per minute to produce an acceleration of 6.8g.

Step-by-step solution

Calculate Speed

We will use the formula for Centripetal Acceleration, which is \(a_c = \frac{v^2}{r}\), where \(a_c\) is the Centripetal Acceleration, \(v\) is the speed, and \(r\) is the radius. We are given \(a_c = 6.8g\) (also needs to be converted to m/s^2), and the radius is \(r = 5.2\) m. We can solve for \(v\) using this formula.

Calculate revolutions per minute

To find the number of revolutions per minute, we first need to find the time it takes for one revolution. This time is called the period, \(T\), and is equal to \(T = \frac{2πr}{v}\). Once we know the period, we can calculate the number of revolutions per minute as \(Revolutions = \frac{60}{T}\).

Key concepts

Centrifuge Motion

Centrifuge motion involves rotating an object around a fixed axis. This motion results in what we call centripetal acceleration. The primary purpose of a centrifuge is often to simulate increased gravitational forces by spinning the object rapidly. The force pushing the object towards the center of rotation is essential in maintaining this motion, allowing us to study various applications like separating substances or training astronauts. In the context of astronaut training, centrifuge motion helps simulate the gravitational forces that astronauts will experience during space travel. The speed at which the centrifuge rotates is crucial, as it determines the magnitude of the centripetal acceleration, thereby replicating different gravitational conditions.

Rotational Dynamics

Rotational dynamics is the study of how forces affect motion in circular paths. It encompasses various aspects such as angular velocity, angular acceleration, and torque. In scenarios like the astronaut centrifuge, understanding rotational dynamics is vital to ensuring the system operates safely and effectively. Key aspects that come into play include:

• Angular Velocity: This is the rate of change of angular position and is often measured in radians per second.
• Angular Acceleration: It describes how quickly the angular velocity changes.
• Torque: The force that causes an object to rotate around an axis.

Grasping these concepts allows engineers to control and predict the motion of rotating systems accurately, ensuring both efficacy and safety.

Centripetal Force

Centripetal force is a crucial component of any rotational motion. It acts perpendicular to the direction of the velocity of the object in motion and points inward towards the center of the rotation. This force is necessary to keep the object moving in a circular path.For an astronaut in a centrifuge:

• Centripetal Force: It must be strong enough to simulate the desired gravitational conditions without being so large as to cause harm.
• Calculation: It can be calculated using the formula: \( F_c = \frac{mv^2}{r} \), where \(m\) is the mass of the object (in this case, the astronaut), \(v\) is the velocity, and \(r\) is the radius of the centrifugal path.

Understanding this force enables precise control over the rotational system, ensuring the safety and success of training simulations.

Astronaut Training

Astronaut training using a centrifuge is designed to prepare astronauts for the extreme conditions of space travel, particularly the intense gravitational forces experienced during launch and re-entry. By training in a centrifuge, astronauts can build tolerance to high g-forces, which is crucial for preventing disorientation or fainting. Key elements of this training:

• Simulates the gravitational pull experienced during critical phases of a mission.
• Helps astronauts understand their physical limits and how to manage them.
• Enables astronauts to practice tasks and maintain operation protocols while subjected to high g-forces.

This training is vital for ensuring the physical preparedness and psychological readiness of astronauts to handle space missions.

Physics Problems

Addressing physics problems, like the one with centrifuges, lays a solid foundation for understanding motion principles in rotational contexts. These problems typically require applying concepts such as centripetal acceleration, force, and speed calculations.For example, solving for speed involves rearranging the centripetal acceleration formula: \( a_c = \frac{v^2}{r} \), allowing us to solve for \(v\) given \(a_c\) and \(r\). Calculating revolutions per minute, meanwhile, involves determining the period of rotation using \( T = \frac{2\pi r}{v} \) and converting it to a frequency-based measure like RPM (revolutions per minute).

• Understanding these foundational principles aids in problem-solving and the application to real-world scenarios.
• Solid grasp of these concepts allows for accurate simulations and predictions in physics experiments and applications.

These problems help deepen our understanding of both basic and complex physical phenomena related to circular motion.