BIO Weighing a Bacterium Scientists are using tiny,...

BIO Weighing a Bacterium Scientists are using tiny, nanoscale cantilevers 4 micrometers long and 500 nanometers wide- essentially miniature diving boards - as a sensitive way to measure mass. An example is shown in FIGURE $13-38 .$ The cantilevers oscillate up and down with a frequency that depends on the mass placed near the tip, and a laser beam is used to measure the frequency. A single $E .$coli bacterium was measured to have a mass of 665 femtograms $=6.65 \times 10^{-16} \mathrm{kg}$ with this device, as the cantilever oscillated with a frequency of 14.5 $\mathrm{MHz}$ . Treating the cantilever as an ideal, massless spring, find its effective force constant.

BIO Weighing a Bacterium Scientists are using tiny, nanoscale
cantilevers 4 micrometers long and 500 nanometers wide-
essentially miniature diving boards - as a sensitive way to measure mass. An example is shown in FIGURE $13-38 .$ The cantilevers
oscillate up and down with a frequency that depends on the mass
placed near the tip, and a laser beam is used to measure the frequency. A single $E .$coli bacterium was measured to have a mass of
665 femtograms $=6.65 \times 10^{-16} \mathrm{kg}$ with this device, as the cantilever oscillated with a frequency of 14.5 $\mathrm{MHz}$ . Treating the cantilever as an ideal, massless spring, find its effective force constant.

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Key Concepts

Simple Harmonic Oscillator

This concept involves systems where a restoring force proportional to displacement leads to sinusoidal oscillations. In such systems, the oscillatory frequency is determined by the interplay between the mass and the spring constant, as described by the equation f = (1/(2?))?(k/m). Understanding simple harmonic motion is crucial for analyzing the dynamic behavior of any mass-spring system.

Spring (Force) Constant

The spring constant, or force constant, quantifies the stiffness of a spring. It relates the force required to displace the spring to the magnitude of that displacement. In the context of oscillatory systems, it directly influences the frequency of vibration, and when the mass and frequency are known, the spring constant can be calculated using the formula k = (2?f)²m.

Cantilever as a Sensing Element

A cantilever is a beam anchored at one end, which can flex in response to applied forces. When used as a sensitive sensor, particularly at the nanoscale, its resonant vibrational frequency changes in response to small mass alterations at its free end. By monitoring these frequency shifts, such devices can be employed to measure very small masses, making them useful in applications like biological sensing.

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