# Frequency of Centrifuge

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1. Apr 9, 2016

### Balsam

1. The problem statement, all variables and given/known data
A research apparatus called a centrifuge undergoes a centripetal acceleration with a magnitude of 3.3x10^6m/s^2. The centrifuge has a radius of 8.4cm. Calculate the frequency of the centrifuge in hertz and in revolutions per minute.

Ac=3.3x10^6,/s^2 r=0.084m

2. Relevant equations
ac=4pi^2rf^2

3. The attempt at a solution
I tried solving this by isolating for f and plugging in all of my given values, but the answer I got was 683963.585Hz while the correct answer is 10 000Hz. Also, how do you convert Hz to revolutions per minute- do you just divide by 60?

2. Apr 9, 2016

### OldYat47

Acceleration = v^2/r. Tangential velocity is the square root of (acceleration X radius), the tangential velocity is 1,665 m / s. The frequency is the tangential velocity divided by the circumference of rotation, or (1,665 m / s) / (5.27 m), or 315.46 cycles per second (Hertz). That gives a frequency of 18,927 cycles per minute.

Let's assume that the correct answer is 10,000 Hz. Then the tangential velocity is 52,779 m / s. The acceleration would be 3.3 *10^9 m / s^2.

ac=4pi^2rf^2 should be ac=4pi^2r/f^2. And by the way, the f is not frequency but time for a single rotation. Check the dimensional analysis.

Last edited: Apr 9, 2016
3. Apr 9, 2016

### drvrm

to find out rev. per minute from rev per sec one should not divide by 60

4. Apr 10, 2016

### OldYat47

Speaking of checking calculations (picture me with a red face), the tangential velocity using a = (v^2 / r) is 1,587.5 m/s. This gives a rate of 300.8 Hz or 18,046.5 RPM.

Using a = (4 * pi^2 * r / t^2) results in a time of 0.00332 seconds per revolution. This agrees at 300.8 Hz, and the tangential velocity agrees at 1,587.5 m/s.

5. Apr 10, 2016

### drvrm

well i am getting about 1000 Hz and if converted to per minute it will be 60 thousand rev per minute but where did i do wrong?

6. Apr 10, 2016

### OldYat47

You may be right, who knows? In any case, here are my numbers and equations.
ac=4pi^2rf^2, above is not correct, should be ac = 4*pi^2*r/f^2. Note that f is not the frequency but the time to complete one revolution. See concurrent dimensional analysis below.
f = [(4*pi^2*r)/ac]^0.5, [(4*9.8696*0.84)/(3*10^6)]^0.5 (meters)/(meters/seconds squared) = (seconds squared)
f=(1.1054*10^-5)^0.5 (seconds squared)^0.5
f=3.325*10^-3 seconds
The frequency is the inverse of that number, or 300.7746 (revolutions (dimensionless) / second).
The tangential velocity is 300.7746 (revolutions per second * meters per revolution), or
(300.7746 * 2 * pi * 0.84) = 1,587.45 (meters/second)

Now let's use ac = v^2/r
v=(ac*r)^0.5, (3*10^6*0.84)^0.5 [(meters/seconds squared)*(meters)]^0.5=(meters squared/seconds squared)^0.5=(meters/second)
v= 1,587.45 (meters/second)

Show your numbers and equations and let's see where the problems lie.

7. Apr 10, 2016

### drvrm

i did it more simply -therefore i might have made errors-
acceleration = w^2 . R where w is angular velocity as w= 2.pi/T where T is time taken to complete one revolution,
in one second no. of rev. n=1 /T so,

w
= 2.pi.n where n is the no. of revolution per sec

acceleration(a) = 4. Pi^2.n^2 . R therefore n= Sqrt ( a / 4.pi^2 .R) ; n= sqrt [ 3.3 . 10^6) /( 4.(3.14)^2 . 0.084) ]
putting in numbers its approx n =1000 this is equivalent to frequency in Hz ,
so in 1 minute i.e. 60 seconds =1000.60 = 60,000 rev. per minute.

however these numbers are very high

Last edited: Apr 10, 2016
8. Apr 11, 2016

### OldYat47

I cannot see what the problem is other than the insistence that n=1,000. Look at your own equation. When I calculate the value of that equation I get 300.7746 Hz, or 18,046.47 RPM. Using any of the other available equations for these numbers the result is the same.

That's all I've got.