We saw in Example 5.6 how a centrifuge can be used to separate cells from a liquid. To inc...

We saw in Example 5.6 how a centrifuge can be used to separate cells from a liquid. To increase the rate at which objects can be separated from solution, it is useful to make the centrifuge’s speed as large as possible. If you want to design a centrifuge of diameter 50 cm to have a force of I06 times the force of Earth’s gravity, what is the speed of the outer edge of the centrifuge? Such a device is called ultraracentrifuge.

Example 5.6

Operation of a Centrifuge

Calculate the settling speed vcell of a cell of radius 10 µm and mass 10−5 mg (= 10−11 kg) in blood using Equation 5.15. Assume the centrifuge has a radius of 10 cm and a rotation rate of 3000 revolutions per minute. How long must the centrifuge run to make the cell settle out?

RECOGNIZE THE PRINCIPLE

Many quantities involved in this problem, such as the radius and mass of a cell, are never known precisely and vary from cell to cell. Moreover, the Stokes’s relation for the drag force, Equation 5.14, applies exactly only for a spherical object, and cells are usually not precisely spherical! Even so, we can use Equation 5.15 to get an approximate value for the speed at which a cell moves. The only unknown quantity in Equation 5.15 is v, the speed of the centrifuge. We can find v from the period of the centrifuge, along with the relation between period and v in Equation 5.1.

SKEdTCH THE PROBLEM

Figure 5.16 shows the problem from the point of view of the cell. According to this noninertial observer, the force of artificial gravity causes an acceleration ac “downward” along the centrifuge tube.

Figure 5.16 Example 5.6. A cell in a centrifuge moves as if there is a force of artificial gravity acting “downward” on the cell.

IDENTIFY THE RELATIONSHIPS AND SOLVE

Our centrifuge makes 3000 revolutions in 1 min, so the time to make 1 revolution is T = 1 min/3000 = 0.020 s, and the circumferential speed is

Inserting this in Equation 5.15 gives

Since the centrifuge tube has a length equal to the radius of the centrifuge (see Fig. 5.14), the cell must travel a distance of at most L = 10 cm = 0.10 m. Using the value found for vcell, the time required for the cell to move this distance is

What does it mean?

A centrifuge with a similar speed is used to separate blood cells from plasma. Allowing for variations in cell size, our analysis shows that it only takes a few seconds to completely separate the cells from the plasma.

Figure 5.14 A centrifuge rotates at an extremely high rate about its axis at C, producing a large centripetal acceleration for the contents of the centrifuge. Here, the contents are a liquid that might contain cells in suspension. The result is that the cells move to the outer end of the tube.