Ch 03: Motion in Two or Three Dimensions

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 03: Motion in Two or Three Dimensions

Problem 29b

Chapter 3, Problem 29b

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?

Verified step by step guidance

First, understand that the acceleration experienced by the astronaut in the centrifuge is due to centripetal acceleration, which is given by the formula: \( a = \frac{v^2}{r} \), where \( v \) is the tangential velocity and \( r \) is the radius of the circular path.

The maximum acceleration experienced by the astronaut's head is given as 12.5g, where \( g \) is the acceleration due to gravity (approximately \( 9.81 \text{ m/s}^2 \)). Therefore, the maximum acceleration \( a_{head} = 12.5 \times 9.81 \text{ m/s}^2 \).

The radius for the head is the full length of the arm, \( r_{head} = 8.84 \text{ m} \). For the feet, the radius is reduced by the height of the astronaut, so \( r_{feet} = 8.84 \text{ m} - 2.00 \text{ m} = 6.84 \text{ m} \).

Using the centripetal acceleration formula, the acceleration at the feet \( a_{feet} \) can be calculated by substituting \( r_{feet} \) into the formula: \( a_{feet} = \frac{v^2}{r_{feet}} \).

Finally, the difference in acceleration between the head and the feet is \( a_{head} - a_{feet} \). Substitute the values of \( a_{head} \) and \( a_{feet} \) to find this difference.

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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 calculated using the formula a = v^2/r, where v is the tangential velocity and r is the radius of the circle. In the context of the centrifuge, this concept helps determine the acceleration experienced by different parts of the astronaut's body.

Gravitational Acceleration (g)

Gravitational acceleration, denoted as g, is the acceleration due to Earth's gravity, approximately 9.81 m/s². In hypergravity conditions, accelerations are expressed as multiples of g, such as 12.5g, indicating the force experienced is 12.5 times that of normal gravity. Understanding this helps quantify the extreme conditions astronauts face in the centrifuge.

Relative Motion in Rotating Systems

Relative motion in rotating systems involves understanding how different points on a rotating body experience different accelerations due to their varying distances from the axis of rotation. In the centrifuge, the astronaut's head and feet are at different radii, causing a difference in centripetal acceleration. This concept is crucial for calculating the acceleration difference between the head and feet.

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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?

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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 in rpm (rev/min) is the arm turning to produce the maximum sustained acceleration?

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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?

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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 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 linear speed of the blade tip, in m/s?

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?

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