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?

Calculate the angular velocity w (in radians per second) using the equation

R*w^2 = 6600 g = 64,700 m/s^2

Use R = 4.28*10^-2 m

w = 1230 rad/s

Convert that to rpm for the answer.

It's about 12,000 rpm.

Well, well, well, talking about spinning blood cells, are we? That's one way to make them really dizzy!

But don't worry, I've got you covered. To find out how many revolutions per minute the sample is making, we'll need to do a bit of math.

First, let's convert the radius from centimeters to meters. So, 4.28 cm becomes 0.0428 meters. Got it? Great!

Now, we know that the centripetal acceleration is 6.60 x 10^3 times larger than the acceleration due to gravity. The acceleration due to gravity on Earth is approximately 9.8 m/s². So, if we multiply that by 6.60 x 10^3, we get the centripetal acceleration.

Now, the centripetal acceleration is given by the formula a = ω²r, where a is the centripetal acceleration, ω is the angular velocity in radians per second, and r is the radius.

Since we want the sample's angular velocity in revolutions per minute, we need to do a little more math. One revolution is equal to 2π radians. So, the angular velocity in radians per second is ω = 2πn, where n is the number of revolutions per minute.

Now, we have everything we need. We can plug in the values and solve for n:

6.60 x 10^3 * 9.8 = (2πn)² * 0.0428

Solving for n, we find that the sample is making approximately 328.5 revolutions per minute.

So, the blood cells in that centrifuge are really spinning their hearts out, I guess! Hope that helps!

To find the number of revolutions per minute the sample is making, we'll need to follow these steps:

Step 1: Identify the given values
- Centripetal acceleration (a) = 6.60 x 10^3 times acceleration due to gravity (g)
- Radius (r) = 4.28 cm

Step 2: Convert the radius from centimeters to meters
We need to convert the radius to meters to maintain consistency with the units of acceleration and for further calculations. There are 100 centimeters in 1 meter, so:

r = 4.28 cm = 4.28 / 100 = 0.0428 m

Step 3: Determine the acceleration due to gravity
The acceleration due to gravity (g) is approximately 9.8 m/s^2.

Step 4: Calculate the centripetal acceleration
Since the centripetal acceleration (a) is given as 6.60 x 10^3 times acceleration due to gravity (g), we can calculate it using the following formula:

a = 6.60 x 10^3 * g
a = 6.60 x 10^3 * 9.8 = 6.468 x 10^4 m/s^2

Step 5: Calculate the angular velocity
The angular velocity (ω) can be calculated using the formula:

a = ω^2 * r
6.468 x 10^4 = ω^2 * 0.0428

Rearranging the equation, we find:

ω^2 = 6.468 x 10^4 / 0.0428

Taking the square root of both sides:

ω = √(6.468 x 10^4 / 0.0428) = 208.9 rad/s

Step 6: Convert the angular velocity from radians per second to revolutions per minute
To convert from radians per second (rad/s) to revolutions per minute (rpm), we can use the following conversion factor:

1 revolution = 2π radians

1 minute = 60 seconds

ω (in rpm) = (ω (in rad/s) * 60) / (2π)

Substituting the values:

ω (in rpm) = (208.9 * 60) / (2π) = 1991.03

Therefore, the sample is making approximately 1991.03 revolutions per minute.

To find the number of revolutions per minute (rpm) that the sample is making, we first need to calculate the centripetal acceleration of the sample.

The centripetal acceleration can be calculated using the formula:

a = ω^2 * r

where:
a is the centripetal acceleration,
ω is the angular velocity (in radians per second),
r is the radius of rotation.

The centripetal acceleration is given as 6.60x10^3 times the acceleration due to gravity (g). So we have:

a = 6.60x10^3 * g

We know that the acceleration due to gravity (g) is approximately 9.8 m/s^2.

Now, we can solve for ω by rearranging the formula:

a = ω^2 * r
ω^2 = a / r
ω = √(a / r)

Substituting the values, we get:

ω = √((6.60x10^3 * g) / r)

The sample is located at a radius of 4.28 cm, which is equivalent to 0.0428 m. Substituting this value, we have:

ω = √((6.60x10^3 * 9.8) / 0.0428)

Calculating this expression, we find ω to be approximately 2,567.60 rad/s.

To convert this angular velocity to revolutions per minute, we multiply it by a conversion factor. There are 2π radians in one revolution and 60 seconds in one minute, so the conversion factor is (2π rad) / (60 s).

rpm = (ω * 60) / (2π)

Substituting the value of ω, we get:

rpm = (2,567.60 * 60) / (2π)

Using a calculator, we can evaluate this expression to find that the sample is making approximately 24,375 rpm.