We saw in Example 5.7 how a centrifuge can be used to separate cells from a liquid. To increase the speed 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 68 cm to have a force of 9 X106 times the force of Earth’s gravity, what is the speed (in rev/s) of the outer edge of the centrifuge? Such a device is called an ultracentrifuge.
To find the speed of the outer edge of the centrifuge, we need to use the concept of centripetal force.
Centripetal force is given by the equation:
F = m * ω² * r
Where:
F is the centripetal force
m is the mass of the object
ω is the angular velocity
r is the radius or distance from the center
In this case, the centripetal force is 9 x 10^6 times the force of Earth's gravity, which we can write as:
F = 9 x 10^6 * g
Where g is the acceleration due to gravity (approximately 9.8 m/s²).
The mass of the object cancels out, so we are left with:
9 x 10^6 * g = ω² * r
Now, we can solve for ω (angular velocity):
ω² = (9 x 10^6 * g) / r
To calculate the speed of the outer edge of the centrifuge, we need to convert the angular velocity to revolutions per second. The angular velocity is the number of revolutions per unit time, so we can use the conversion:
ω = 2π * f
Where:
ω is the angular velocity in radians per second
f is the frequency in revolutions per second
Now, substitute ω in terms of f:
(2π * f)² = (9 x 10^6 * g) / r
Solving for f:
f = sqrt((9 x 10^6 * g) / (4π² * r))
Now, let's plug in the given values:
g = 9.8 m/s²
r = 68 cm = 0.68 m
π ≈ 3.14159
f = sqrt((9 x 10^6 * 9.8) / (4 * 3.14159² * 0.68))
Using a calculator, evaluate the expression inside the square root and then take the square root of the result:
f ≈ sqrt(28203975.86 / 9.2167) ≈ sqrt(3066326.85) ≈ 1751.39 rev/s
Therefore, the speed of the outer edge of the centrifuge is approximately 1751.39 revolutions per second.