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Chapter 10: Problem 18

(I) Pilots can be tested for the stresses of flying high-speed jets in a whirling "human centrifuge," which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed. (a) What was its angular acceleration (assumed constant), and \((b)\) what was its final angular speed in rpm?

Short Answer

Expert verified
Angular acceleration is \(\frac{2\pi}{90} \text{ rad/s}^2\) and final speed is 20 rpm.

Step by step solution

01

Understand the problem

We are given a centrifuge that completes 20 revolutions in 1 minute and we need to find its angular acceleration and final angular speed assuming constant acceleration. We will use the formulas for rotational kinematics to find (a) the angular acceleration and (b) the final angular speed in revolutions per minute (rpm).
02

Convert revolutions to radians

Since angular motion is typically described in radians, we convert the number of revolutions to radians. Given: 20 revolutions. Since 1 revolution equals \(2\pi\) radians: \[ \text{Total angular displacement } \theta = 20 \times 2\pi = 40\pi \text{ radians}. \]
03

Apply the rotational kinematic equation for angular acceleration

We use the equation for angular displacement \( \theta = \omega_0 t + \frac{1}{2}\alpha t^2 \), where \(\omega_0\) is the initial angular velocity (0 in this case), \(\alpha\) is the angular acceleration, and \(t\) is the time.\[ 40\pi = 0 + \frac{1}{2}\alpha (60^2). \]Solving for \(\alpha\),\[ \alpha = \frac{80\pi}{3600} = \frac{2\pi}{90} \text{ rad/s}^2. \]
04

Calculate the final angular speed in rad/s

We use the equation \( \omega = \omega_0 + \alpha t \) to find the final angular velocity \(\omega\), where \(\omega_0 = 0\) and \(\alpha\) was calculated in Step 3.\[ \omega = 0 + \left(\frac{2\pi}{90}\right) \times 60 = \frac{120\pi}{90} \text{ rad/s} = \frac{4\pi}{3} \text{ rad/s}. \]
05

Convert the final angular speed to rpm

Finally, convert the angular velocity from rad/s to revolutions per minute (rpm).1 revolution = \(2\pi\) radians, and there are 60 seconds per minute:\[ \omega_{\text{rpm}} = \left( \frac{4\pi}{3} \right) \times \left( \frac{1 \, \text{rev}}{2\pi \, \text{rad}} \right) \times 60 = \frac{120}{6} = 20 \, \text{rpm}. \]

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Key Concepts

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

Angular Acceleration
Angular acceleration is an important concept in rotational kinematics. It describes how the rate of rotation of an object changes over time. In the context of the "human centrifuge" problem, we assume a constant angular acceleration, meaning that the rate at which the centrifuge spins is increasing uniformly during the minute it takes to complete 20 revolutions.
  • Angular acceleration is symbolically represented by \( \alpha \).
  • Its unit of measure is radians per second squared (rad/s2).
  • When calculating angular acceleration, if the object starts from rest \((\omega_0 = 0)\) and reaches a certain angular displacement \( \theta \), we use the equation: \[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \] for our calculations, solving for \( \alpha \) typically involves algebraic manipulation.
In our exercise, the total angular displacement is \( 40\pi \) radians over 60 seconds. This leads us to find that the angular acceleration \( \alpha \) is \( \frac{2\pi}{90} \) rad/s虏, which tells us how quickly the angular velocity of the centrifuge is increasing second to second.
Angular Velocity
Angular velocity is the rate at which an object rotates or spins. In our exercise, we are interested in finding out the final angular velocity of the centrifuge after it completes the 20 revolutions.
  • The symbol for angular velocity is usually \( \omega \).
  • It's expressed in units like radians per second (rad/s), but often we convert to rpm (revolutions per minute) for practical purposes.
  • The calculation involves using the formula \[ \omega = \omega_0 + \alpha t \] where \( \alpha \) is the angular acceleration and \( t \) is time.
In the exercise, we determined the final angular velocity in radians per second first by calculating \( \omega = \frac{4\pi}{3} \) rad/s. Converting this to revolutions per minute by using the relation that 1 revolution is \( 2\pi \) radians leads us to conclude that the final angular velocity is 20 rpm. This means that after 1 minute, the centrifuge is spinning 20 full revolutions per minute.
Angular Displacement
Angular displacement is the angle through which an object moves on a circular path. It represents the change in the position of the object and is a crucial part of rotational motion analysis.
  • It's symbolized by \( \theta \).
  • This measurement is typically in radians, though it can also be expressed in revolutions or degrees.
  • One full revolution equals \( 2\pi \) radians.
In our problem, the centrifuge makes 20 complete revolutions. To convert this to radians, we multiply the number of revolutions by \( 2\pi \), resulting in a total angular displacement of \( 40\pi \) radians. Understanding this concept is fundamental as it allows us to relate other quantities like angular velocity and acceleration through different kinematic equations. In essence, angular displacement links how much the object has rotated from its start to its finish during the event.

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Most popular questions from this chapter

Bicycle gears: \((a)\) How is the angular velocity \(\omega_{\mathrm{R}}\) of the rear wheel of a bicycle related to the angular velocity \(\omega_{\mathrm{F}}\) of the front sprocket and pedals? Let \(N_{\mathrm{F}}\) and \(N_{\mathrm{R}}\) be the number of teeth on the front and rear sprockets, respectively, Fig. \(10-64\) The teeth are spaced the same on both sprockets and the rear sprocket is firmly attached to the rear wheel. \((b)\) Evaluate the ratio \(\omega_{\mathrm{R}} / \omega_{\mathrm{F}}\) when the front and rear sprockets (1) have 52 and 13 teeth, respectively, and (c) when they have 42 Rear sprocket \(\quad\) and 28 teeth.

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