It has recently become possible to \"weigh\" DNA molecules by measuring the infl

Question

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 kg/m3) 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 13M attached to a spring. (The factor of 13 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 10 MHz . Attachment of a DNA molecule caused the frequency to decrease by 55 Hz . (Figure 1)

Part A: What was the mass of the DNA?

Explanation / Answer

The spring-mass system vibrates with frequency f, corresponding to an angular velocity =2f, given by:

= sqrt(M/k)
f = /(2) = sqrt(M / (3k)) / (2)

Where k is the spring constant, and the (1/3) inside the square root accounts for the effective mass M/3 of the modeled spring-mass system. This gives two equations:

f = sqrt(M/k) / (2)
f + f = sqrt((M + M)/k) / (2)

We know f and f, but M, M and k are unknown. However, we can estimate M+M to high accuracy as follows.

Take the differential to estimate the effect of a small mass increment:

d = [(1/2) / sqrt(M / (3k))] * dM/ (3k)
d = [(1/2) * sqrt(M / (3k)) * 3k/M] * dM / (3k) = (1/2) * dM/M
d/ = df/f = (1/2) dM/M

So M/M ~= dM/m = 2df/f ~= f/f, to about as many significant digits as we know f/f itself.

This means that M+M is approximately 2*M*(10 MHz + 55Hz)/(10 MHz).

M+M=(2*2300*(10^7+55))/(10^7)=4600kg

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