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Problem 15.58 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 M attached to a spring. (The factor of 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 11 MHz. Attachment of a DNA molecule caused the frequency to decrease by 45 Hz. (Figure 1) Figure 4000 nm 1400 nm Thickness 100 nm = 1 of 1 Part A What was the mass of the DNA? Express your answer with the appropriate units. MDNA3.68 10²¹ ● kg Provide Feedback www. Submit Previous Answers Request Answer * Incorrect; Try Again; 4 attempts remaining ?
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Transcribed Image Text:Problem 15.58 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 M attached to a spring. (The factor of 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 11 MHz. Attachment of a DNA molecule caused the frequency to decrease by 45 Hz. (Figure 1) Figure 4000 nm 1400 nm Thickness 100 nm = 1 of 1 Part A What was the mass of the DNA? Express your answer with the appropriate units. MDNA3.68 10²¹ ● kg Provide Feedback www. Submit Previous Answers Request Answer * Incorrect; Try Again; 4 attempts remaining ?
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