It has recently become possible to weigh DNA molecules by measuring the influence of their mass on a nano-oscillator. FIGURE P15.54 shows a thin rectangular

It has recently become possible to €œweigh€ DNA molecules by measuring the influence of their mass on a nano-oscillator. FIGURE P15.54 shows a thin rectangular cantilever etched out of silicon (density 2300 kg/m³) with a small gold dot (not visible) at the end. If pulled down and released, the end of the cantilever vibrates with SHM, 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 1/3 M attached to a spring. (The factor of 1/3 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 12 MHz. Attachment of a DNA molecule caused the frequency to decrease by 50 Hz. What was the mass of the DNA?

4000 nm [ 400 nm Thickness = 100 nm FIGURE P15.54