How Does Mass Affect the Resonant Frequency of a Quartz-Crystal Monitor?

[hadoque]

Homework Statement

Model the vibrating quartz-crystal thickness monitor as a mass(m)-spring combination, where k is the spring constant.
a) What is the resonant frequency?
b) Show that as additional mass $$\delta m$$ deposits the the difference in resonant frequency or frequency shift is given by $$\Delta f \approx f_0 \delta m /2m$$ for $$\delta m / m \ll 1$$

Homework Equations

The Attempt at a Solution

a) F=-kx
$$f = \frac{1}{2 \pi} \sqrt{\frac{k}{m}}$$

b) $$\Delta f = \frac{1}{2 \pi} \sqrt{\frac{k}{\delta m +m}} - f_0$$

Is this equation the right approach? Seems difficult to eliminate one whole term on the right side, and the k, how to get rid of that?

 

[tiny-tim]

hi hadoque!

can't you just differentiate 1/√m ?

 

[hadoque]

Wow, why didn't I think about that? I've doing to little math lately:)

 

[tiny-tim]

relax …

do what i do …

swim around and stare at things! ​

(you are a haddock, aren't you?)​

 

[hadoque]

I sure am, and that was a good advice!