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 12 MHz . Attachment of a DNA molecule caused the frequency to decrease by 54 Hz .

Part A: What was the mass of the DNA?

Answer 1

Mass of cantilever,

M = $\rho$ V

M = 2300*(40000*10^-9*400*10^-9*100*10^-9)

M = 3.68*10^-16 kg

mass as a block of mass 13 M attached to a spring,

M1 = 13* 3.68*10^-16 = 4.784*10^-15 kg

Let, mass of DNA = m

We know that,

f $\propto$ 1 / $\sqrt{M}$

f1 / f2 = sqrt (M2 / M1)

f1 / (f1 - 54) = sqrt (M1 + m / M1)

[f1 / (f1 - 54)]^2 = (M1 + m / M1)

m = M1 * [(f1 / (f1 - 54))^2 - 1]

m =  4.784*10^-15 * [(12*10⁶ / (12*10⁶ - 54))^2 - 1]

m =4.784*10^-15 * 0.0000090

m = 4.305*10^-20 kg