twanging a ruler

[maxmax99]

Hi everyone
on your project idea 'Measuring Vibrational Frequency with Light' in the introduction it says:
"Have you ever "twanged" a ruler sticking off the edge of a desk and then pulled it back to hear the funny sound it makes? When you twang the ruler, it vibrates at a natural frequency determined by the stiffness of the ruler material and the length of the ruler that is able to vibrate. As you pull back on the ruler, the length that is free to vibrate becomes shorter and shorter. The frequency of the vibration increases, and you hear that funny, rising sound."
that is exactly what i have done. what i need now is a formula relating frequency, stiffness and length.
i know f=kx, and am looking for a formula similar to this.
any help would be great. thanks very much

[deleted-2131]

maxmax99,

The equation you have provided, f(x) = kx is, unless I am misunderstanding you, Hooke's Law. Hooke's law is an equation that gives the force needed to stretch a spring a certain distance. The force, in newtons, is f(x), the distance, in meters, is x, and k is an experimentally determined spring constant. I'm not exactly sure what kind of a formula you are trying to find that is similar to Hooke's Law. If I understand your question, you are trying to figure out an equation that relates the frequency at which the ruler is vibrating, the length of the ruler that is vibrating, and the stiffness of the stiffness of the ruler. Is this correct?

Please post back with more information *as much detail as possible* so that we can better understand your question and help you.

Looking forward to your reply!

[jasonzimmerman]

The f(x) = kx that you are referring to is Hooke's Law, which when combined with Newton's second law can be derived into a simple vibration equation for a simple spring/mass system where the first mode of natural frequency of vibration (assuming no damping) can be described by the equation in the link below (sorry, this describes it as well as I could and it is already done):

http://en.wikipedia.org/wiki/Vibration# ... ut_damping

What you are looking for is a bit more complicated and requires some derivation from the simple answer above. The answer is actually also listed on wikipedia in the form of a tuning fork frequency calculation and this assumes that your ruler is basically rectangular in its shape (cross-section) and uniform from one end to the other. As the cross-sectional shape of the ruler (or whatever thing you are analyzing) changes, the formula is not as simple and I can further provide details if you are interested. Here is the link:

http://en.wikipedia.org/wiki/Tuning_fork

The modulus for wood (assuming you have a wood ruler?) is also there if you click on "Young's Modulus" if you actually want to go through the calculation and compare against your experimental results.

By reading your original request, this formula will relate the 3 things you requested. You should notice that changing the length is much more effective than changing the stiffness. The only variable I would add to your list that is pretty important is density of the material. Materials vary in stiffness, but the stiffness and density are not necessarily always related. The best example I can give is copper vs steel. Copper is actually more dense (more weight per unit of volume), but if you have ever played with copper wire, you know it is not very "stiff" (lower Young's Modulus or sometimes referred to as "Modulus of elasticity") compared to steel.

I hope this helps - great question!!!

[deleted-2574]

Hi maxmax99,

Another reference extending the responses above is to look at Problem 6 at: http://galileo.phys.virginia.edu/classe ... swers.html

[maxmax99]

thankyou all for your quick replies!
In repsonse to your question terik daly, yes you are quite right, those are the variables i was looking to combine - although now it seems it is a bit more complicated than that!
And david kallman - thanks for that link. i think i might extend my project to cover one of those fields if i have time.
jasonzimmerman, thankyou so much, your post was great. the tuning fork formula is perfect - just what i needed. i used rulers made of plastic, wood and metal; so now it is simply a case of measuring the area, modulus and density of the rulers.
i think that i now know everything i need, but i will post again if i have some problems and feel free to post with tips or advice or extensions or anything like that!

[deleted-2574]

Hi maxmax99,

You're welcome! It's always good to hear that we're helping. Thanks for your post.

I just reviewed my post, and saw how it could be confusing. I was referring to the second problem 6 (in Part II - Short Answers) rather than the first problem 6 (in Part I - Multiple Choice Questions).

[jasonzimmerman]

maxmax99,

It's great to hear that the responses were helpful and you are on your way!!

Best wishes!