A centrifuge is a device in which a small container of material is rotated at a high speed on a circular path. Such a device is used in medical laboratories, for instance, to cause the more dense red blood cells to settle through the less dense blood serum and collect at the bottom of the container. Suppose the centripetal acceleration of the sample is 5.65 103 times as large as the acceleration due to gravity. How many revolutions per minute is the sample making, if it is located at a radius of 4.80 cm from the axis of rotation?
To find the number of revolutions per minute that the sample is making, we need to determine the angular velocity of the rotation.
First, let's find the centripetal acceleration (ac) using the formula:
ac = R * ω^2
where ac is the centripetal acceleration, R is the radius of rotation, and ω is the angular velocity.
Given that the centripetal acceleration is 5.65 * 10^3 times as large as the acceleration due to gravity (g), we have:
ac = 5.65 * 10^3 * g
Now, let's rearrange the equation to solve for ω:
ω = √(ac / R)
Substituting the given values:
R = 4.80 cm = 0.0480 m
ac = 5.65 * 10^3 * g
Before substituting for ac, we need to convert the acceleration due to gravity from m/s^2 to g, where 1 g = 9.8 m/s^2.
ac = 5.65 * 10^3 * 9.8 m/s^2 = 5.527 * 10^4 m/s^2
Now we can substitute the values into the equation:
ω = √((5.527 * 10^4 m/s^2) / 0.0480 m)
Simplifying this gives:
ω = 1.444 * 10^3 rad/s
Finally, let's convert the angular velocity to revolutions per minute. Since there are 2π radians in one revolution, we can use the conversion factor:
1 revolution = 2π radians
1 minute = 60 seconds
ω (in revolutions per minute) = (1.444 * 10^3 rad/s) * (1 revolution / 2π radians) * (60 seconds / 1 minute)
Calculating this gives:
ω ≈ 13795.92 revolutions per minute
Therefore, the sample is making approximately 13796 revolutions per minute.