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 5.15 x 10^3 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 3.63 cm from the axis of rotation?
To find the number of revolutions per minute (rpm) that the sample is making, we can use the following steps:
Step 1: Calculate the centripetal acceleration of the sample.
Given that the centripetal acceleration is 5.15 x 10^3 times the acceleration due to gravity, we can express this as:
a_c = 5.15 x 10^3 * g
where g is the acceleration due to gravity, which is approximately 9.8 m/s^2.
Step 2: Convert the radius from centimeters to meters.
The radius is given as 3.63 cm. To use this value in the calculations, it needs to be converted to meters. We divide it by 100:
r = 3.63 cm / 100 = 0.0363 m
Step 3: Calculate the angular velocity.
The centripetal acceleration can be expressed as:
a_c = ω^2 * r
where ω is the angular velocity.
Rearranging the formula, we get:
ω = sqrt(a_c / r)
Substituting the known values, we have:
ω = sqrt((5.15 x 10^3 * g) / 0.0363)
Step 4: Convert angular velocity to revolutions per minute.
The angular velocity is usually measured in radians per second (rad/s), but we want the answer in rpm. There are 2π radians in one revolution, and 60 seconds in one minute. So, to convert ω to rpm, we use the formula:
rpm = (ω * 60) / (2π)
Substituting the value of ω into the formula, we get:
rpm = ((sqrt((5.15 x 10^3 * g) / 0.0363)) * 60) / (2π)
Now we can calculate the value using a calculator or symbolic algebra system.
After performing the calculations, the number of revolutions per minute (rpm) the sample is making can be determined.