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Ch 03: Motion in Two or Three Dimensions
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoYoung & Freedman Calc 14th EditionCh 03: Motion in Two or Three DimensionsProblema 29c
Capitolo 3, Problema 29c

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 12.5g. How fast in rpm (rev/min) is the arm turning to produce the maximum sustained acceleration?

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First, understand that the problem involves centripetal acceleration, which is the acceleration experienced by an object moving in a circle at constant speed. The formula for centripetal acceleration \( a_c \) is \( a_c = \frac{v^2}{r} \), where \( v \) is the tangential speed and \( r \) is the radius of the circle.
Given that the maximum sustained acceleration is 12.5g, where \( g \) is the acceleration due to gravity (approximately 9.81 m/s²), calculate the maximum centripetal acceleration: \( a_c = 12.5 \times 9.81 \text{ m/s}^2 \).
The radius \( r \) of the centrifuge is given as 8.84 m. Use the centripetal acceleration formula \( a_c = \frac{v^2}{r} \) to solve for the tangential speed \( v \): \( v = \sqrt{a_c \times r} \).
Once you have the tangential speed \( v \), convert this speed into angular speed \( \omega \) using the relation \( \omega = \frac{v}{r} \).
Finally, convert the angular speed \( \omega \) from radians per second to revolutions per minute (rpm). Use the conversion factor: \( 1 \text{ rad/s} = \frac{60}{2\pi} \text{ rpm} \).

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Centripetal Acceleration

Centripetal acceleration is the acceleration experienced by an object moving in a circular path, directed towards the center of the circle. It is given by the formula a_c = v^2/r, where v is the tangential speed and r is the radius of the circle. In the context of the centrifuge, this acceleration is what the astronaut experiences as 'hypergravity'.
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Conversion of Units

To solve the problem, it's essential to convert the given acceleration from 'g' units to meters per second squared (m/s²), where 1g = 9.81 m/s². Additionally, converting the final angular speed from radians per second to revolutions per minute (rpm) is necessary for the answer, using the conversion factors 1 revolution = 2π radians and 1 minute = 60 seconds.
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Relationship Between Linear and Angular Velocity

The relationship between linear velocity (v) and angular velocity (ω) in circular motion is given by v = ωr, where r is the radius. This relationship helps in determining the angular speed of the centrifuge arm, which is crucial for calculating the revolutions per minute needed to achieve the specified centripetal acceleration.
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Domanda del libro di testo

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 12.5g. How fast must the astronaut's head be moving to experience this maximum acceleration?

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Domanda del libro di testo

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 12.5g. What is the difference between the acceleration of his head and feet if the astronaut is 2.00 m tall?

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