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 2.28 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 2.97 cm from the axis of rotation?

Ac = v^2/R = omega^2 R

omega in radians/second
Ac = 9.81*2.28*10^3 (good grief!)

omega^2 = 9.81*2.28*10^3/.0297

omega = 868 radians /second

868 rad/s (1rev/2pi rad)(60s/min)
= 8287 rpm

Well, aren't you just spinning me around with this question! Let's break it down, shall we?

First, we need to find the centripetal acceleration of the sample. We know that it's 2.28 x 10^3 times the acceleration due to gravity. Now, since the acceleration due to gravity is approximately 9.8 m/s^2, we can simply multiply that by 2.28 x 10^3 to get the centripetal acceleration.

Next, we'll use the formula for centripetal acceleration, which is given by a = ω^2 * r, where ω represents the angular velocity and r is the radius. We can rearrange this formula to solve for ω: ω = sqrt(a / r).

Now, since we're dealing with revolutions per minute, we need to convert our units. There are 60 seconds in a minute and 2π radians in 1 revolution. So, we can convert ω to revolutions per minute by using the following conversion factor: 1 revolution / (2π radians) = 1 minute / (60 seconds).

Finally, we can plug in the given values to find the angular velocity, ω, and then convert it to revolutions per minute. So, buckle up and let's crunch those numbers!

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

The centripetal acceleration (ac) is given by the equation:
ac = ω^2 * r
where ω is the angular velocity and r is the radius.

We are given that the centripetal acceleration is 2.28 x 10^3 times as large as the acceleration due to gravity (g). Therefore,
ac = 2.28 x 10^3 * g

We know that the acceleration due to gravity is approximately 9.8 m/s^2 or 9.8 x 10^3 cm/s^2.

So, ac = 2.28 x 10^3 * 9.8 x 10^3 cm/s^2
= 22.344 x 10^6 cm/s^2

Next, we need to calculate the angular velocity ω using the centripetal acceleration ac.

ω = ac / r

Substituting the values, we get:
ω = 22.344 x 10^6 cm/s^2 / (2.97 cm)
= 7.52 x 10^6 s^-2

Finally, to find the number of revolutions per minute, we need to convert the angular velocity from rad/s to rpm.

1 revolution = 2π radians
1 minute = 60 seconds

Therefore,
angular velocity (ω) in rpm = (ω in rad/s) * (60 sec/min) / (2π rad/rev)

Plugging in the values, we have:
ω in rpm = (7.52 x 10^6 s^-2) * (60 sec/min) / (2π rad/rev)
≈ 22.70 x 10^6 rpm (rounded to two decimal places)

Thus, the sample is making approximately 22.70 x 10^6 revolutions per minute.

To find the number of revolutions per minute a sample is making on a centrifuge, we can use the relationship between centripetal acceleration, radius, and angular velocity.

The centripetal acceleration (ac) is given as 2.28 x 10^3 times the acceleration due to gravity (g). Mathematically, we can write:

ac = (2.28 x 10^3) * g

The centripetal acceleration is related to the angular velocity (ω) and radius (r) by the equation:

ac = ω^2 * r

Now, let's solve for ω. Rearranging the equation, we have:

ω = √(ac/r)

Substituting the given values for ac and r, we have:

ω = √[((2.28 x 10^3) * g) / (2.97 cm)]

Next, we need to convert the radius to meters, since g is given in m/s^2:

r = 2.97 cm = 0.0297 m

Substituting the radius into the equation:

ω = √[((2.28 x 10^3) * g) / (0.0297 m)]

We can now calculate the value of ω.

Once we have the angular velocity, we can convert it to revolutions per minute. Since one revolution is equivalent to a complete circle (2π radians), the conversion factor is:

1 revolution = 2π radians

Similarly, one minute is equal to 60 seconds. Therefore, we can write:

1 minute = 60 seconds

To convert ω to revolutions per minute, we multiply it by the conversion factor:

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

Now, we can calculate the number of revolutions per minute the sample is making by substituting the calculated ω into the equation.

Finally, we can solve for the result, giving the number of revolutions per minute the sample is making on the centrifuge.