Skip to main content
Ch 03: Motion in Two or Three Dimensions
Young & Freedman Calc - University Physics 15th Edition
Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook
All textbooksYoung & Freedman Calc 15th EditionCh 03: Motion in Two or Three DimensionsProblem 33c
Chapter 3, Problem 33c

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?

Verified step by step guidance
1
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: \( 1 \text{ rad/s} = \frac{60}{2\pi} \text{ rpm} \).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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'.
Recommended video:
Guided course
06:48
Intro to Centripetal Forces

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.
Recommended video:
Guided course
07:46
Unit Conversions

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.
Recommended video:
Guided course
11:10
Converting Between Linear & Rotational
Related Practice
Textbook 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 12.5g. How fast must the astronaut's head be moving to experience this maximum acceleration?

1
views
Textbook 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 12.5g. What is the difference between the acceleration of his head and feet if the astronaut is 2.00 m tall?

8
views
Textbook Question

A railroad flatcar is traveling to the right at a speed of 13.0 m/s relative to an observer standing on the ground. Someone is riding a motor scooter on the flatcar (Fig. E3.30). What is the velocity (magnitude and direction) of the scooter relative to the flatcar if the scooter's velocity relative to the observer on the ground is 18.0 m/s to the right?

3
views
Textbook Question

A model of a helicopter rotor has four blades, each 3.40 m long from the central shaft to the blade tip. The model is rotated in a wind tunnel at 550 rev/min. What is the radial acceleration of the blade tip expressed as a multiple of g?

Textbook Question

A 'moving sidewalk' in an airport terminal moves at 1.0 m/s and is 35.0 m long. If a woman steps on at one end and walks at 1.5 m/s relative to the moving sidewalk, how much time does it take her to reach the opposite end if she walks In the opposite direction?

3
views
Textbook Question

A 'moving sidewalk' in an airport terminal moves at 1.0 m/s and is 35.0 m long. If a woman steps on at one end and walks at 1.5 m/s relative to the moving sidewalk, how much time does it take her to reach the opposite end if she walks in the same direction the sidewalk is moving?

2
views