Problem 29 - University Physics with Modern Physics

Question

At its Ames Research Center, NASA uses its large "20-G" centrifuge to test the effects of very large accelerations ("hypergravity") on test pilots and astronauts. In this device, an arm 8.84 m long rotates about one end in a horizontal plane, and an astronaut is strapped in at the other end. Suppose that he is aligned along the centrifuge's arm with his head at the outermost end. The maximum sustained acceleration to which humans are subjected in this device is typically 12.5\(g\). (a) How fast must the astronaut's head be moving to experience this maximum acceleration? (b) What is the \(difference\) between the acceleration of his head and feet if the astronaut is 2.00 m tall? (c) How fast in rpm (rev/min) is the arm turning to produce the maximum sustained acceleration?

Step-by-Step Solution

Answer

(a) 33.05 m/s, (b) 36.9 m/s², (c) 35.75 rpm.

Step 1: Understanding the Centrifugal Acceleration

In a rotating system, the centrifugal acceleration is given by \( a = \frac{v^2}{r} \), where \( v \) is the linear velocity and \( r \) is the radius of rotation. We need to find \( v \) for the acceleration \( a = 12.5g \), and \( g \) is the acceleration due to gravity, approximately \( 9.81 \, \text{m/s}^2 \).

Step 2: Calculate the Velocity at Maximum Acceleration

Use the formula for centrifugal acceleration: \( a = \frac{v^2}{r} \). Set \( a = 12.5g = 12.5 \times 9.81 \, \text{m/s}^2 \) and \( r = 8.84 \, \text{m} \) (length of the arm). Solve for \( v \):\[ 12.5 \times 9.81 = \frac{v^2}{8.84} \]\[ v^2 = 12.5 \times 9.81 \times 8.84 \]\[ v = \sqrt{12.5 \times 9.81 \times 8.84} \]\[ v \approx 33.05 \, \text{m/s} \]

Step 3: Calculate Acceleration Difference Between Head and Feet

For his feet, the radius is \( r = 8.84 \, \text{m} - 2.00 \, \text{m} = 6.84 \, \text{m} \). Calculate acceleration at the feet:\[ a_{\text{feet}} = \frac{v^2}{6.84} \]Substitute \( v = 33.05 \, \text{m/s} \) calculated earlier:\[ a_{\text{feet}} = \frac{(33.05)^2}{6.84} \approx 159.8 \, \text{m/s}^2 \]Difference in acceleration:\[ \Delta a = (12.5 \times 9.81) - 159.8 \approx 36.9 \, \text{m/s}^2 \]

Step 4: Calculate the Rotational Speed in RPM

The relation between linear velocity \( v \), angular velocity \( \omega \), and radius is \( v = r \omega \). Calculate \( \omega \):\[ \omega = \frac{v}{r} = \frac{33.05}{8.84} \approx 3.74 \, \text{rad/s} \]Convert \( \omega \) from rad/s to rpm using the conversion factor \( 1 \, \text{rpm} = \frac{2\pi}{60} \, \text{rad/s} \):\[ \omega_{\text{rpm}} = \frac{3.74 \times 60}{2\pi} \approx 35.75 \, \text{rpm} \]

Key Concepts

Hypergravity

Hypergravity refers to conditions where the force of gravity experienced is greater than the force of Earth’s gravity. This condition is tested using devices like a large centrifuge at NASA's Ames Research Center.
The concept is crucial for understanding the physical strain astronauts undergo when exposed to acceleration forces much greater than those encountered on Earth. In hypergravity scenarios, the body has to manage forces many times the standard acceleration due to gravity, or "g".

• The limit of human tolerance in such conditions is carefully tested.
• The experiences gained help develop suitable support structures for astronauts.

Knowing about hypergravity is important for ensuring astronauts can safely travel and work in different gravitational environments.

Centrifuge Physics

Centrifuge physics involves understanding how rotation can create artificial gravity through centrifugal force. A centrifuge is a device that spins an object around a central axis, applying outward force.
In practical terms, NASA uses a centrifuge to mimic the extreme forces of space travel. The astronaut is positioned at the far end, simulating high-gravitational forces as the long arm spins.

• The arm length impacts the radius of rotation, crucial in angular velocity calculations.
• The linear velocity of the astronaut's head determines the experienced acceleration.

These setups allow determination of safe limits and effects of hypergravity on the human body.

Rotational Motion Calculations

Rotational motion calculations are key to understanding the physics behind centrifuges. Specifically, they involve determining the velocity and acceleration an individual experiences.
Key equations enable calculations for different aspects of rotational motion:

• Centrifugal acceleration formula: \[a = \frac{v^2}{r}\]
• The angular velocity \( \omega \): \[\omega = \frac{v}{r}\]

To convert angular speed to revolutions per minute (rpm), use:\[ \omega_{\text{rpm}} = \frac{\omega \times 60}{2\pi} \]
These calculations help understand specific details, like how fast a centrifuge must rotate to reach a desired acceleration. This knowledge is essential for designing experiments and safely conducting tests under hypergravity conditions.