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?
To determine the number of revolutions per minute the sample is making, we need to find the angular velocity of the sample first.
We know that the centripetal acceleration (ac) of the sample is 6.60x10^3 times as large as the acceleration due to gravity (g). Mathematically, we can express this as:
ac = 6.60 x 10^3 * g
The centripetal acceleration is given by the formula:
ac = ω^2 * r
Where:
ω is the angular velocity (in radians per second)
r is the radius of rotation (in meters)
We are given the radius of rotation as 4.28 cm, which we convert to meters:
r = 4.28 cm = 0.0428 m
Substituting the values into the formula, we get:
6.60 x 10^3 * g = ω^2 * 0.0428
Next, we rearrange the equation to solve for ω:
ω = √(6.60 x 10^3 * g / 0.0428)
To find the number of revolutions per minute, we need to convert the angular velocity from radians per second to revolutions per minute.
1 revolution = 2π radians
1 minute = 60 seconds
So, to convert from radians per second (ω) to revolutions per minute (rpm), we use the following conversion factor:
1 rpm = (1 revolution / 2π radians) * (60 seconds / 1 minute)
Finally, we substitute the value of ω and simplify the expression to find the number of revolutions per minute.