/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 The National Aeronautics and Spa... [FREE SOLUTION] | 91Ó°ÊÓ

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The National Aeronautics and Space Administration (NASA) studies the physiological effects of large accelerations on astronauts. Some of these studies use a machine known as a centrifuge. This machine consists of a long arm, to one end of which is attached a chamber in which the astronaut sits. The other end of the arm is connected to an axis about which the arm and chamber can be rotated. The astronaut moves on a circular path, much like a model airplane flying in a circle on a guideline. The chamber is located \(15 \mathrm{~m}\) from the center of the circle. At what speed must the chamber move so that an astronaut is subjected to 7.5 times the acceleration due to gravity?

Short Answer

Expert verified
The chamber must move at a speed of approximately 33.2 m/s.

Step by step solution

01

Understand the Problem

The problem is asking for the speed at which a chamber, located 15 meters from the center of a circle, must move to make an astronaut experience an acceleration of 7.5 times the acceleration due to gravity.
02

Determine the Required Acceleration

The acceleration due to gravity is approximately 9.8 m/s². Therefore, 7.5 times this acceleration is: \[a = 7.5 \times 9.8 = 73.5 \text{ m/s}^2\]
03

Recall the Formula for Centripetal Acceleration

The formula for the centripetal acceleration is given by \(a = \frac{v^2}{r}\), where \(v\) is the tangential speed, and \(r\) is the radius of the circle.
04

Substitute the Known Values

We know that \(a = 73.5 \text{ m/s}^2\) and \(r = 15 \text{ m}\). Substituting these into the formula, we have: \[73.5 = \frac{v^2}{15}\]Multiply both sides by 15 to solve for \(v^2\): \[73.5 \times 15 = v^2\]
05

Calculate the Speed

Calculate the value of \(v^2\): \[v^2 = 1102.5\]Take the square root to find \(v\): \[v = \sqrt{1102.5}\]Using a calculator, find that: \[v \approx 33.2 \text{ m/s}\]
06

Verify the Solution

Check our calculations to ensure no errors have been made: - The acceleration required matches the calculated acceleration when using the formula for centripetal acceleration. - The computed speed value makes physical sense given the constraints described in the problem.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Centrifuge
A centrifuge is a machine that rotates around a fixed axis, creating an outward force due to its spin. Imagine swinging a bucket filled with water over your head. As you spin, the water stays inside due to this perceived outward force. That's similar to what happens in a centrifuge. The key purpose of a centrifuge in astronaut simulations is to generate high acceleration levels that mimic the forces experienced during rocket launch or reentry.
  • It consists of an arm attached to a rotating axis.
  • At one end of the arm, a chamber holds the test subject or object.
  • By rotating at high speeds, the centrifuge can simulate conditions of gravity much stronger than Earth’s.
This tool is crucial for NASA when testing how astronauts deal with these intense forces before their missions. Understanding how a centrifuge operates helps you comprehend how artificial gravity or acceleration is produced in controlled environments.
Astronaut Acceleration Studies
In the realm of astronautics, studying acceleration is vital to understanding the effects on the human body. Astronauts undergo immense accelerations when being launched into space or when re-entering Earth's atmosphere. These accelerations can be multiple times the force of gravity experienced on Earth.
  • The simulations help determine the physical limits of astronauts.
  • Using machines like a centrifuge, researchers can safely study physiological responses.
  • These studies ensure astronauts are fit to withstand the stresses of space travel.
Through controlled experiments mimicking various gravitational forces, scientists gather essential data, contributing to safe and successful space missions.
Circular Motion
Circular motion is a fundamental concept in physics, describing any object that moves along a circular path. When an astronaut is in a centrifuge, they experience circular motion. Understanding the principles involved is crucial in exploring how objects interact under different forces.
  • Objects moving in a circle have a constant changing direction.
  • The centripetal force is necessary to keep the object in its circular path.
  • This force points towards the center of the circle, enabling circular motion to occur.
In the context of the centrifuge problem, knowing these dynamics aids in solving for variables like speed or radius, ensuring a comprehensive understanding of any rotational system.
Physics Problem Solving
Solving physics problems involves understanding the question, identifying known values, using appropriate formulas, and checking if the solution makes sense. When tackling a physics problem regarding circular motion, like with a centrifuge, here's a simple plan to follow:
  • First, translate the problem into variables and known quantities.
  • Use relevant equations - for instance, the centripetal acceleration formula: \(a = \frac{v^2}{r}\).
  • Substitute known values and simplify to find the unknown variable.
  • Cross-verify calculations to ensure physical plausibility.
By systematically addressing problems, students enhance their logical reasoning, allowing them to tackle a wide range of scientific questions effectively.

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Most popular questions from this chapter

Concept Questions Interactive LearningWare 5.2 at illustrates good problemsolving techniques for this type of problem. Two cars are traveling at the same speed of \(27 \mathrm{~m} / \mathrm{s}\) on a curve that has a radius of \(120 \mathrm{~m}\). Car \(\mathrm{A}\) has a mass of \(1100 \mathrm{~kg},\) and car \(\mathrm{B}\) has a mass of \(1600 \mathrm{~kg}\). Without doing any calculations, decide (a) which car, if either, has the greater centripetal acceleration and (b) which car, if either, experiences the greater centripetal force. Justify your answers.

Before attempting this problem, review Examples 7 and 8 and 7 and 8 . Two curves on a highway have the same radii. However, one is unbanked and the other is banked at an angle \(\theta\). A car can safely travel along the unbanked curve at a maximum speed \(v_{0}\) under conditions when the coefficient of static friction between the tires and the road is \(\mu_{\mathrm{s}}\). The banked curve is frictionless, and the car can negotiate it at the same maximum speed \(v_{0}\). Find the angle \(\theta\) of the banked curve.

At amusement parks, there is a popular ride where the floor of a rotating cylindrical room falls away, leaving the backs of the riders "plastered" against the wall. Suppose the radius of the room is \(3.30 \mathrm{~m}\) and the speed of the wall is \(10.0 \mathrm{~m} / \mathrm{s}\) when the floor falls away. (a) What is the source of the centripetal force acting on the riders? (b) How much centripetal force acts on a \(55.0-\mathrm{kg}\) rider? (c) What is the minimum coefficient of static friction that must exist between a rider's back and the wall, if the rider is to remain in place when the floor drops away?

A satellite has a mass of \(5850 \mathrm{~kg}\) and is in a circular orbit \(4.1 \times 10^{5} \mathrm{~m}\) above the surface of a planet. The period of the orbit is two hours. The radius of the planet is \(4.15 \times 10^{6} \mathrm{~m} .\) What is the true weight of the satellite when it is at rest on the planet's surface?

How long does it take a plane, traveling at a constant speed of \(110 \mathrm{~m} / \mathrm{s}\), to fly once around a circle whose radius is \(2850 \mathrm{~m} ?\)

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