A centrifuge is a device in which small container of material is rotated at a high speed on a circular path. Such a device is a 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 6.60x10^3 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 4.28 cm from the axis of rotation?

Question

A centrifuge is a device in which small container of material is rotated at a high speed on a circular path. Such a device is a 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 6.60x10^3 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 4.28 cm from the axis of rotation?

Answer 1

To find the number of revolutions per minute the sample is making, we need to first find the angular velocity of the sample, and then convert it to revolutions per minute.

Let's start by finding the angular velocity using the formula:

centripetal acceleration = radius × angular velocity^2

The centripetal acceleration is given as 6.60x10^3 times the acceleration due to gravity, which we can represent as:

centripetal acceleration = 6.60x10^3 * g

Where g represents the acceleration due to gravity (9.8 m/s^2).

The radius is given as 4.28 cm, which we should convert to meters:

radius = 4.28 cm = 0.0428 m

Now we can substitute these values into the formula:

6.60x10^3 * g = 0.0428 m * angular velocity^2

Solving for angular velocity:

angular velocity^2 = (6.60x10^3 * g) / (0.0428 m)

angular velocity = √((6.60x10^3 * g) / (0.0428 m))

Next, let's convert the angular velocity to revolutions per minute. One revolution is equal to 2π radians.

To convert from radians per second to revolutions per minute, we'll use the following conversion factors:

1 revolution = 2π radians
1 minute = 60 seconds

First, we need to find the number of radians per second:

radians per second = angular velocity

Now, let's find the number of revolutions per minute:

revolutions per minute = (radians per second / (2π)) * (60 seconds / 1 minute)

Substituting the value of radians per second:

revolutions per minute = (angular velocity / (2π)) * (60 seconds / 1 minute)

Now we can substitute the value of angular velocity we calculated earlier into the formula and solve for revolutions per minute.