# Dimensional analysis – spring mass systems

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

1. Spring supported from above (you can do is horizontally but friction can be a problem).
2. A series of equal masses that can be hung from the end of the spring alone or together.
3. 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.ka.mb.gc= T-1 Ma.T-2a Mb Lc.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

1. 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.
2. Add a second mass and repeat the frequency measurement.
3. 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/n1/2. 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.