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 6.91 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 5.90 cm from the axis of rotation?

To solve this problem, we first need to determine the centripetal acceleration of the sample.

Given:
Centripetal acceleration (a_c) = 6.91 x 10^3 g
Radius (r) = 5.90 cm = 0.0590 m

Acceleration due to gravity (g) = 9.8 m/s^2

We know that centripetal acceleration is given by the formula:

a_c = (v^2) / r

Where:
v is the linear velocity (in m/s)
r is the radius (in m)

We can rewrite the formula as:

v = sqrt(a_c * r)

Substituting the given values, we get:

v = sqrt((6.91 x 10^3) * 9.8 * 0.0590)

Now let's calculate the value of v:

v = sqrt(405.4884) = 20.136 m/s

Next, we need to convert the linear velocity to angular velocity (ω), which is measured in radians per second. The relationship between linear velocity and angular velocity is given by:

v = ω * r

Solving for ω:

ω = v / r = 20.136 / 0.0590 = 341.98 rad/s

Finally, we need to convert the angular velocity to revolutions per minute (RPM). One revolution corresponds to 2π radians. The conversion factor is:

1 revolution = 2π radians

To find the RPM, we divide the angular velocity by 2π and then convert seconds to minutes:

RPM = (ω / 2π) * 60 = (341.98 / (2π)) * 60 = 3249.38 RPM

Therefore, the sample is making approximately 3249.38 revolutions per minute.

To find the number of revolutions per minute the sample is making, we need to use the concept of centripetal acceleration and relate it to rotational motion.

First, let's calculate the centripetal acceleration of the sample using the formula:

ac = (v^2) / r

where ac is the centripetal acceleration, v is the linear velocity, and r is the radius of the circular path.

In this case, we are given that the centripetal acceleration is 6.91 x 10^3 times the acceleration due to gravity (g).

So, ac = 6.91 x 10^3 * g

Next, let's relate the linear velocity (v) to the number of revolutions per minute.

The linear velocity can be calculated by using the formula:

v = 2πr / t

where v is the linear velocity, r is the radius of the circular path, and t is the time taken for one revolution.

To find the time taken for one revolution (t), we can use the relationship:

t = 1 / (number of revolutions per minute)

Then, we can substitute the value of t back into the formula for linear velocity:

v = 2πr * (number of revolutions per minute)

Finally, we can substitute the value of v into the formula for centripetal acceleration:

ac = (4π^2r^2 * (number of revolutions per minute)^2) / r

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

6.91 x 10^3 * g = (4π^2 * r * (number of revolutions per minute)^2) / r

Simplifying the equation:

(number of revolutions per minute)^2 = (6.91 x 10^3 * g * r) / (4π^2)

Taking the square root of both sides:

number of revolutions per minute = √[(6.91 x 10^3 * g * r) / (4π^2)]

Now we can plug in the values. Given that the radius (r) is 5.90 cm (or 0.0590 m) and the acceleration due to gravity (g) is 9.81 m/s^2:

number of revolutions per minute = √[(6.91 x 10^3 * 9.81 * 0.0590) / (4π^2)]

Evaluating this expression will give you the number of revolutions per minute the sample is making.

v^2 / r = 6.91E3 * g

rpm = 60 [v / (2 π r)]