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 5.27 x 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 2.50 cm from the axis of rotation?

Question

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 5.27 x 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 2.50 cm from the axis of rotation?

Answer 1

To find the number of revolutions per minute (RPM) that the sample is making, we need to use the formula for centripetal acceleration and relate it to the acceleration due to gravity.

The formula for centripetal acceleration is given by:

a = (v^2) / r

Where:
- a is the centripetal acceleration,
- v is the linear velocity of the object,
- r is the radius of its circular path.

Since we know that the centripetal acceleration is 5.27 x 10^3 times larger than the acceleration due to gravity (g), we can set up the equation:

a = (5.27 x 10^3) * g

Rearranging the equation to solve for v:

v = sqrt(a * r)

Now, we can plug in the values. The radius is given as 2.50 cm, which we need to convert to meters to be consistent with the acceleration due to gravity. 1 cm is equal to 0.01 meters, so:

r = 2.50 cm * 0.01 m/cm = 0.025 m

We can approximate the acceleration due to gravity as 9.8 m/s^2. Plugging in the values, we have:

v = sqrt((5.27 x 10^3) * (9.8 m/s^2) * (0.025 m))

Simplifying the equation further:

v = sqrt(1318.75) m/s

v ≈ 36.27 m/s

Finally, to find the number of revolutions per minute, we need to convert the linear velocity (v) to angular velocity (ω) using the formula:

ω = v / r

Substituting the values:

ω = (36.27 m/s) / (0.025 m)

ω ≈ 1450.8 rad/s

To convert this to revolutions per minute, we multiply ω by (60 s / 2π rad) and divide by 1 revolution:

RPM = (ω * (60 s / 2π rad)) / (1 rev)

RPM ≈ (1450.8 rad/s * (60 s / 2π rad)) / (1 rev)

RPM ≈ 13811.29 rev/min

Therefore, the sample is making approximately 13811.29 revolutions per minute.