# Difficult problem about Simple Harmonic Motion using only mass to frequency ratios

## Homework Statement

It has recently become possible to "weigh" DNA molecules by measuring the influence of their mass on a nano-oscillator. Figure shows a thin rectangular cantilever etched out of silicon (density 2300 {\rm kg/m^3}) with a small gold dot at the end. If pulled down and released, the end of the cantilever vibrates with simple harmonic motion, moving up and down like a diving board after a jump. When bathed with DNA molecules whose ends have been modified to bind with gold, one or more molecules may attach to the gold dot. The addition of their mass causes a very slight-but measurable-decrease in the oscillation frequency. A vibrating cantilever of mass M can be modeled as a block of mass {\textstyle{1 \over 3}}\,M attached to a spring. (The factor of {\textstyle{1 \over 3}} arises from the moment of inertia of a bar pivoted at one end.) Neither the mass nor the spring constant can be determined very accurately-perhaps to only two significant figures-but the oscillation frequency can be measured with very high precision simply by counting the oscillations. In one experiment, the cantilever was initially vibrating at exactly 13 {\rm MHz}. Attachment of a DNA molecule caused the frequency to decrease by 57 {\rm Hz}.

What was the mass of the DNA?

The dimensions of the beam are 4000nm, 400nm and 100nm

## Homework Equations

w = 2$$\pi$$f
f = w/2$$\pi$$

## The Attempt at a Solution

I calculated the mass of the beam to be 3.68 x 10-16 kg. I worked the f = w/2$$\pi$$ equation to get f= sqrt(M/k)/2$$\pi$$ and then f + $$\Delta$$f = sqrt[(M+$$\Delta$$M)/k]/2$$\pi$$.

Then I took the derivative of w = sqrt(M/3k) to get dw = 0.5w(dM/M) after simplification. I just don't know where to go with it. I know I need to set up a ratio but I just can't put it together in my head. Any ideas?

Any help is greatly appreciated!

## The Attempt at a Solution

ideasrule
Homework Helper

Try to calculate df/dM and express the result in terms of the initial frequency and mass.

How do I do that though? I calculated df/dM from f = sqrt
(M/k)/2pi to get (1/4kpi)*sqrt(M/k)*(k/M) but I don't know where to go with this. Did I do the right thing? I know I'm missing a key piece in my understanding but I can't figure out the connection.