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Chapter 3: Problem 64

Commercial ultracentrifuges can rotate at rates of \(100,000 \mathrm{rpm}\) (revolutions per minute). As a consequence, they can create accelerations on the order of \(800,000 \mathrm{~g}\). (A " \(g\) " represents an acceleration of \(9.8 \mathrm{~m} / \mathrm{s}^{2}\).) Find the distance from the rotation axis of the sample chamber in such a device. Calculate the speed of an object traveling under the given conditions.

Short Answer

Expert verified
The distance from the rotation axis, or radius, of the sample chamber and the speed of an object traveling under these conditions can be found using the conversions and formulas from these steps.

Step by step solution

01

Convert units

The acceleration is given in units of g. Convert this to m/s^2 using the value of g provided: \(800,000 \, g = 800,000 \times 9.8 \, m/s^2\). The rotation rate is given in revolutions per minute. Convert this to radian per second using the relation 1 rev=2蟺 rad and 1 minute=60 seconds. Therefore, \(100,000 \, rpm = 100,000 \times \frac{2\pi}{60} \, rad/s\).
02

Calculate radius

Use the formula for centripetal acceleration which is \(a = \omega^2 r\), where \(a\) is the centripetal acceleration, \(\omega\) is the angular velocity and \(r\) is the radius. Here, we need to solve for \(r\) so we're rearranging it to: \(r = \frac{a}{\omega^2}\). Plug in the values from step 1 and calculate the radius.
03

Calculate speed

The speed of the object can be calculated using the formula \(v = \omega r\). Input the values of \(\omega\) and \(r\) (calculated in steps 1 and 2) into the equation to find the speed.

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Key Concepts

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

Ultracentrifuge
An ultracentrifuge is a high-speed laboratory device that spins samples at extremely high speeds. This device uses the principles of centripetal force and angular velocity to separate components within a sample. It can reach speeds of up to 100,000 revolutions per minute (rpm), which is essential for applications such as molecular biology, biochemistry, and even in the analysis of large molecules like proteins.

  • These high speeds generate enormous centripetal acceleration, often exceeding hundreds of thousands of times the force of gravity on Earth ("g"), allowing the ultracentrifuge to effectively separate small particles based on density.
  • Key applications include separating plasma from blood, isolating proteins or DNA, and purifying viruses.
Understanding how an ultracentrifuge works is crucial for fields that require precise separation of complex mixtures.
Angular Velocity
Angular velocity is crucial for understanding the workings of devices like ultracentrifuges. It measures how fast an object rotates about an axis and is expressed in radians per second.

  • To convert from revolutions per minute (rpm) to radians per second, remember that one revolution equals 2蟺 radians, and you must divide by 60 to convert minutes to seconds. For example, 100,000 rpm becomes: \( 100,000 \times \frac{2\pi}{60} \ rad/s\).
  • This conversion is vital since many rotational dynamics equations require angular velocity in radians per second.
In ultracentrifuge calculations, angular velocity helps determine both the radius of rotation and the speed of materials spun in the device.
Centripetal Force
Centripetal force is the key force acting on a rotating object, directed towards the center of the circle. This force is necessary to keep an object moving in a circular path, rather than a straight line. The magnitude of centripetal force can be calculated with the formula: \( F_c = m \cdot a = m \cdot \omega^2 \cdot r\), where:
  • \( m \) is the mass of the object
  • \( \omega \) is the angular velocity
  • \( r \) is the radius of the circle.
Centripetal acceleration, a component of this force, is what enables an ultracentrifuge to achieve high separation efficiency by exerting forces many times greater than gravity. This acceleration causes denser particles to move outward more rapidly, facilitating the separation process. Understanding centripetal force and acceleration is essential for interpreting how ultracentrifuges operate at such high speeds.

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