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Chapter 10: Problem 22

A centrifuge is a common laboratory instrument that separates components of differing densities in solution. This is accomplished by spinning a sample around in a circle with a large angular speed. Suppose that after a centrifuge in a medical laboratory is turned off, it continues to rotate with a constant angular deceleration for 10.2 s before coming to rest. (a) If its initial angular speed was 3850 rpm, what is the magnitude of its angular deceleration? (b) How many revolutions did the centrifuge complete after being turned off?

Short Answer

Expert verified
(a) 39.60 rad/s虏. (b) 328 revolutions.

Step by step solution

01

Convert Initial Angular Speed

The initial angular speed is given in rotations per minute (rpm), but we'll need it in radians per second for calculations. First, convert 3850 rpm to revolutions per second:\[\frac{3850 \text{ revolutions}}{1 \text{ minute}} \times \frac{1 \text{ minute}}{60 \text{ seconds}} = \frac{3850}{60} \text{ revolutions per second} \approx 64.17 \text{ rps}\]Next, convert revolutions per second to radians per second:\[64.17 \text{ rps} \times 2\pi \text{ radians per revolution} \approx 403.92 \text{ rad/s}\]Thus, the initial angular speed \( \omega_0 = 403.92 \text{ rad/s} \).
02

Determine Angular Deceleration

The centrifuge comes to rest (final angular speed, \( \omega_f = 0 \text{ rad/s} \)) with a constant angular deceleration over 10.2 seconds. Using the equation:\[\omega_f = \omega_0 + \alpha \cdot t\]where \( \alpha \) is the angular deceleration, solve:\[0 = 403.92 + \alpha \cdot 10.2\]\[\alpha = -\frac{403.92}{10.2} \approx -39.60 \text{ rad/s}^2\]The magnitude of angular deceleration is \(|\alpha| = 39.60 \text{ rad/s}^2\).
03

Calculate Total Revolutions

To find the total number of revolutions the centrifuge makes after being turned off, use the equation for angular displacement \( \theta \):\[\theta = \omega_0 \cdot t + \frac{1}{2} \cdot \alpha \cdot t^2\]Substitute the known values:\[\theta = 403.92 \cdot 10.2 + \frac{1}{2} \cdot (-39.60) \cdot (10.2^2)\]\[\theta = 4119.38 - 2057.04 \approx 2062.34 \text{ radians}\]Convert radians to revolutions:\[\frac{2062.34 \text{ radians}}{2\pi \text{ radians per revolution}} \approx 328.31 \text{ revolutions}\]Thus, the centrifuge makes approximately 328 revolutions after being turned off.

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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 fascinating tool commonly used in labs to separate particles of different densities within a solution. This is achieved by spinning the sample rapidly in a circle, much like how a washing machine spins clothes to remove water.
When the centrifuge rotates, the denser particles move outward towards the edge, while the lighter components stay closer to the center. This process of separation leverages the principle of centripetal force.
  • In medical labs, centrifuges are vital for tasks like separating plasma from blood.
  • They offer precise control over rotation speed, ensuring efficiency in separation.
When a centrifuge is turned off, it doesn't stop immediately. Due to its high angular speed, it takes time to come to a halt, experiencing a phase called angular deceleration, which we'll explore next.
Radians per Second
Radians per second (rad/s) is a unit of angular velocity, which describes how quickly something is rotating. It's much like miles per hour or kilometers per hour but for circular motion.
To make calculations in physics easier, especially those involving circular motion, angular speeds are often converted into radians per second. This conversion allows for consistent and comprehensive calculations.
  • Consider that 1 revolution corresponds to an angle of 2蟺 radians, so converting from revolutions per minute (rpm) to radians per second is a crucial step.
  • For example, if a centrifuge spins at 3850 rpm, converting this to rad/s is essential to calculate further variables, like angular displacement.
By using radians per second, we can effectively work through formulas to determine other aspects of motion, like the angular deceleration of our centrifuge.
Angular Displacement
Angular displacement represents the angle, in radians, through which an object rotates. It's a crucial concept in determining how much a rotating object has turned, similar to how linear displacement tells you how far an object has traveled in a straight line.
In the context of our centrifuge, the total angular displacement tells us how many radians it spins through during deceleration.
  • This involves using the kinematic equation for rotational motion: \( \theta = \omega_0 \cdot t + \frac{1}{2} \cdot \alpha \cdot t^2 \).
  • Substituting values can tell us not just the "distance" in terms of angular motion but also helps calculate the total number of revolutions.
Understanding angular displacement is integral for analyzing how long and how far the centrifuge spins before coming to rest.
Revolutions per Minute
Revolutions per minute (rpm) is a standard unit for measuring the speed of rotation. It indicates how many full turns are completed in one minute and is widely used in devices that operate with rotating parts.
In applications involving angular motion, like our centrifuge, rpm provides an initial reference point for how fast the object is turning.
  • To perform physics calculations, rpm is typically converted into radians per second.
  • This conversion allows us to calculate other quantities such as angular velocity or displacement effectively.
Thus, when a centrifuge is turned off from an initial speed of 3850 rpm, understanding its rpm helps us start accurately informing other dependent calculations, such as angular deceleration and displacement.

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Most popular questions from this chapter

Suppose a bicycle wheel is rotated about an axis through its rim and parallel to its axle. (a) Is its moment of inertia about this axis greater than, less than, or equal to its moment of inertia about its axle? (b) Choose the best explanation from among the following: I. The moment of inertia is greatest when an object is rotated about its center. II. The mass and shape of the wheel remain the same. III. Mass is farther from the axis when the wheel is rotated about the rim.

A ceiling fan is rotating at 0.96 rev /s. When turned off, it slows uniformly to a stop in \(2.4 \mathrm{min}\). (a) How many revolutions does the fan make in this time? (b) Using the result from part (a), find the number of revolutions the fan must make for its speed to decrease from 0.96 rev \(/ \mathrm{s}\) to \(0.48 \mathrm{rev} / \mathrm{s}\) .

The Earth's rate of rotation is constantly decreasing, causing the day to increase in duration. In the year 2006 the Earth took about 0.840 s longer to complete 365 revolutions than it did in the year \(1906 .\) What was the average angular acceleration of the Earth during this time? Give your answer in \(\mathrm{rad} / \mathrm{s}^{2}\).

A lawn mower has a flat, rod-shaped stecl blade that rotates about its center. The mass of the blade is \(0.65 \mathrm{kg}\) and its length is 0.55 m. (a) What is the rotational energy of the blade at its operating angular speed of 3500 rpm? (b) If all of the rotational kinetic energy of the blade could be converted to gravitational potential energy, to what height would the blade rise?

A compact disk (CD) speeds up uniformly from rest to 310 rpm in \(3.3 \mathrm{s}\). (a) Describe a strategy that allows you to calculate the number of revolutions the CD makes in this time. (b) Use your strategy to find the number of revolutions.
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