You don’t need fluid mechanics to demonstrate the use of dimensional analysis to a fluids class. This demonstration uses a dimensional analysis to examine the relationship between frequency of oscillation of a spring and the mass supported by the spring.

Equipment

- Spring supported from above (you can do is horizontally but friction can be a problem).
- A series of equal masses that can be hung from the end of the spring alone or together.
- A stopwatch.

Analysis

I tend to do the analysis first for any dimensional analysis demonstration and then test it with the physical demonstration. The goal is to find the functional relationship between the spring-mass frequency (*f*) and the possible controlling parameters, namely the spring constant (*k*), mass (*m*), and acceleration due to gravity (*g*). That is, we seek

*f=f(k,m,g)*

writing out each parameters dimensions we get

[*f*]=T^{-1}, [*k*]=M.T^{-2}, [*m*]=M, [*g*]=L.T^{-2}

There are 4 parameters and 3 independent dimensions so we can get 1 non-dimensional group. We can, therefore, write

P=*f*.*k*^{a}.*m*^{b}.*g*^{c}= T^{-1} M^{a}.T^{-2a} M^{b} L^{c}.T^{-2c}

Collecting powers of L, M, and T leads to a set of linear equations

(T) 0=-1-2a-2c (M) 0=a+b (L) 0=c

The solution to this set of equations is

a=-1/2 and b=1/2

That is

P=*f(m/k)*^{1/2}

As there is only one P it must be a constant (say *C*) and

*f=C(k/m)*^{1/2}

Therefore, increasing *m* will decrease the frequency *f.* This can be seen in the demonstration

Demonstration

- Place one of the masses on the end of the spring, pull it down and release it. Estimate the frequency by counting how many oscillation there are in 30 seconds.
- Add a second mass and repeat the frequency measurement.
- Keep adding mass and measuring frequency until you are out of mass.

Taking the first mass as being 1 the second as 2, third 3, etc. (assuming all the masses added were equal) then you can use the dimensional analysis and the first measured frequency to make a prediction about the frequencies of the larger mass systems. I ran this in my office and measured the following frequencies.

1 – 2.5 Hz 2 – 1.9 Hz 3- 1.6 Hz

Using the unit mass as the standard then for a mass of *n* the frequency will be *2.5/n*^{1/2}. This leads to frequency predictions of 2 – 1.8 Hz, and 3 – 1.5 Hz. These are both quite close to the measured frequencies.

Thanks to Abdul Khan for lending me his spring – mass system.

An index of all the demonstrations posted on this blog can be found here. Don’t forget to follow @nbkaye on twitter for updates to this blog. If you have a demonstration that you use in class that you would like to share on this blog please email me (nbkaye@clemson.edu). I also welcome comments (through the comments section or via email) on improving the demonstrations.

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