It has recently become possible to 'weigh' DNA molecules by measuring the influence of their mass on a nano-oscillator. FIGURE P15.58 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 ⅓M attached to a spring. (The factor of ⅓ 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?

Verified step by step guidance

1. Start by recalling the formula for the frequency of a simple harmonic oscillator: \( f = \frac{1}{2\pi} \sqrt{\frac{k}{m}} \), where \( f \) is the frequency, \( k \) is the spring constant, and \( m \) is the effective mass of the system. For the cantilever, the effective mass is \( \frac{1}{3}M \), where \( M \) is the total mass of the cantilever.
2. When the DNA molecule attaches to the cantilever, its mass adds to the effective mass of the system. Let the mass of the DNA molecule be \( m_{\text{DNA}} \). The new effective mass becomes \( \frac{1}{3}M + m_{\text{DNA}} \). The new frequency is given as \( f' = \frac{1}{2\pi} \sqrt{\frac{k}{\frac{1}{3}M + m_{\text{DNA}}}} \).
3. The problem states that the initial frequency \( f \) is 12 MHz (\( 12 \times 10^6 \) Hz) and the new frequency \( f' \) is 50 Hz less, so \( f' = 12 \times 10^6 - 50 \). Substitute these values into the frequency formula to set up two equations: one for \( f \) and one for \( f' \).
4. Divide the equation for \( f' \) by the equation for \( f \) to eliminate the spring constant \( k \). This gives \( \frac{f'}{f} = \sqrt{\frac{\frac{1}{3}M}{\frac{1}{3}M + m_{\text{DNA}}}} \). Square both sides to simplify: \( \left(\frac{f'}{f}\right)^2 = \frac{\frac{1}{3}M}{\frac{1}{3}M + m_{\text{DNA}}} \).
5. Rearrange the equation to solve for \( m_{\text{DNA}} \): \( m_{\text{DNA}} = \frac{\frac{1}{3}M}{\left(\frac{f}{f'}\right)^2 - 1} \). Substitute the known values for \( f \), \( f' \), and \( M \) (if provided or estimated) to calculate the mass of the DNA molecule.

Key Concepts

Simple Harmonic Motion (SHM)

Simple Harmonic Motion refers to the oscillatory motion of an object where the restoring force is directly proportional to the displacement from its equilibrium position. In the context of the cantilever, it means that as the cantilever vibrates, it experiences forces that pull it back towards its rest position, resulting in a periodic motion characterized by a specific frequency.

Frequency and Mass Relationship

The frequency of oscillation in a system like the cantilever is inversely related to the square root of its mass. When mass is added to the system, such as when DNA molecules attach to the cantilever, the overall mass increases, leading to a decrease in the oscillation frequency. This relationship can be quantitatively described using formulas derived from the principles of SHM.

Measurement of Oscillation Frequency

In experiments involving oscillating systems, the frequency of oscillation can be measured with high precision, often by counting the number of oscillations over a set period. This precise measurement allows for the detection of small changes in frequency, which can be correlated to changes in mass, such as the addition of DNA molecules to the cantilever, enabling the determination of their mass.