Problem 43 - College Physics

Question

\(\bullet\) \(\bullet\) Weighing a virus. In February \(2004,\) scientists at Purdue University used a highly sensitive technique to measure the mass of a vaccinia virus (the kind used in smallpox vaccine). The procedure involved measuring the frequency of oscillation of a tiny sliver of silicon (just 30 nm long) with a laser, first without the virus and then after the virus had attached itself to the silicon. The difference in mass caused a change in the frequency. We can model such a process as a mass on a spring. (a) Show that the ratio of the frequency with the virus attached \(\left(f_{\mathrm{s}+\mathrm{v}}\right)\) to the frequency without the virus \(\left(f_{\mathrm{s}}\right)\) is given by the formula $$\frac{f_{\mathrm{S}+\mathrm{V}}}{f_{\mathrm{S}}}=\frac{1}{\sqrt{1+\frac{m_{\mathrm{v}}}{m_{\mathrm{S}}}}},$$ where \(m_{\mathrm{v}}\) is the mass of the virus and \(m_{\mathrm{s}}\) is the mass of the silicon sliver. Notice that it is not necessary to know or measure the force constant of the spring. (b) In some data, the silicon sliver has a mass of \(2.10 \times 10^{-16} \mathrm{g}\) and a frequency of \(2.00 \times 10^{15} \mathrm{Hz}\) without the virus and \(2.87 \times 10^{14} \mathrm{Hz}\) with the virus. What is the mass of the virus, in grams and femtograms?

Step-by-Step Solution

Answer

The mass of the virus is \(9.98 \times 10^{-15}\) grams or 9.98 femtograms.

Step 1: Understand the problem

We need to find a relationship between the frequencies with and without the virus attached, using the given formula, and then calculate the mass of the virus when actual frequency values are given for both conditions.

Step 2: Model the system mathematically

When considering a mass on a spring, the frequency of oscillation is given by \(f = \frac{1}{2\pi} \sqrt{\frac{k}{m}}\) where \(k\) is the force constant and \(m\) is the mass. When a virus attaches to the silicon sliver, the effective mass becomes \(m_s + m_v\).

Step 3: Find frequency ratio

By writing the expression for frequency with and without the virus, we have \(f_s = \frac{1}{2\pi} \sqrt{\frac{k}{m_s}}\) and \(f_{s+v} = \frac{1}{2\pi} \sqrt{\frac{k}{m_s + m_v}}\). The ratio \(\frac{f_{s+v}}{f_s}\) becomes \(\frac{\sqrt{\frac{k}{m_s + m_v}}}{\sqrt{\frac{k}{m_s}}}\).

Step 4: Simplify the frequency ratio

Take the ratio \(\frac{\sqrt{\frac{k}{m_s + m_v}}}{\sqrt{\frac{k}{m_s}}}\) which simplifies to \(\sqrt{\frac{m_s}{m_s + m_v}}\). This is \(\frac{1}{\sqrt{1 + \frac{m_v}{m_s}}}\), as required.

Step 5: Substitute frequency values and solve for virus mass

Use the formula \(\frac{f_{s+v}}{f_s} = \frac{1}{\sqrt{1 + \frac{m_v}{m_s}}}\). Substitute \(f_{s+v} = 2.87 \times 10^{14} \text{ Hz}\) and \(f_s = 2.00 \times 10^{15} \text{ Hz}\), and solve for \(m_v\).

Step 6: Calculate the frequency ratio

First calculate \(\frac{f_{s+v}}{f_s}\) which is \(\frac{2.87 \times 10^{14}}{2.00 \times 10^{15}} = 0.1435\).

Step 7: Rearrange and solve for virus mass

Set up the equation \(\frac{1}{\sqrt{1 + \frac{m_v}{2.10 \times 10^{-16}}}} = 0.1435\). Solve for \(m_v\). This involves squaring both sides and rearranging terms to isolate \(m_v\).

Step 8: Calculate the mass of the virus

Square both sides: \((0.1435)^2 = \frac{1}{1 + \frac{m_v}{2.10 \times 10^{-16}}}\). This gives \(0.0206 = \frac{1}{1 + \frac{m_v}{2.10 \times 10^{-16}}}\). Rearranging gives \(1 + \frac{m_v}{2.10 \times 10^{-16}} = \frac{1}{0.0206}\) or \(48.54\). Therefore, \(\frac{m_v}{2.10 \times 10^{-16}} = 47.54\). Multiply both sides by \(2.10 \times 10^{-16}\) to find \(m_v\).

Step 9: Convert virus mass to femtograms

After calculations, \(m_v = 9.98 \times 10^{-15} \text{ grams}\). Converting grams to femtograms: \(1 \text{ gram} = 10^{15} \text{ femtograms}\), thus \(m_v = 9.98 \text{ femtograms}\).

Key Concepts

Mass-Spring System

In physics, a mass-spring system is a common model used to describe oscillatory motion. Imagine a weight attached to a spring that can stretch and compress. This system can move back and forth in a regular manner, which we call oscillation. The frequency of this oscillation depends on two main factors: the stiffness of the spring (known technically as the spring constant, or "k") and the mass attached to the spring.

The basic formula for the frequency (\(f\)) of oscillation is \(f = \frac{1}{2\pi} \sqrt{\frac{k}{m}}\). Here, \(k\) represents the force constant of the spring, and \(m\) represents the mass. This formula shows that as the mass increases, the oscillation frequency decreases, and vice versa. This relationship is crucial for understanding how additional masses, like viruses, attached to the silicon sliver can alter the oscillation frequency in nanotechnology applications.

Virus Mass Measurement

Measuring the mass of something as tiny as a virus is a challenging task, but it's possible with advanced techniques. One method involves using a tiny piece of silicon known as a nano-sliver. In the problem we're addressing, a laser is used to measure the frequency of oscillation of this sliver. First, they measure the frequency without the virus. Then, they measure it again after a virus has landed on the sliver.

The mass of the virus causes a change in the frequency, which we can calculate using the mass-spring system model. This method allows scientists to estimate the mass of very small objects indirectly by observing how they affect the motion of a known system. It's similar to adding a bit of weight to a guitar string and noticing how it changes the pitch of the note it produces.

Frequency Ratio Calculation

One of the key steps to determining the virus's mass involves calculating the frequency ratio. This ratio compares the frequency of the silicon sliver's oscillation with the virus attached, \(f_{s+v}\), to its frequency without the virus, \(f_s\). The given formula for this ratio is:

• \[\frac{f_{s+v}}{f_s} = \frac{1}{\sqrt{1 + \frac{m_v}{m_s}}}\]

This formula tells us how much the frequency changes due to the virus's mass \(m_v\) when the mass of the silicon \(m_s\) is known.
By manipulating this equation, you can isolate the virus's mass \(m_v\), thus making it possible to deduce how the virus impacts the system. This calculation is foundational to many scientific fields, including virology and materials science, where understanding the behavior of tiny masses is crucial.

Silicon Nano-Sliver

The silicon nano-sliver used in the measurement process is exceptionally small, measuring just 30 nanometers in length. This minuscule size is what allows it to be affected by the mass of a virus, which would be negligible if a larger object were used. Silicon is chosen due to its favorable mechanical and electronic properties.

In this exercise, the silicon sliver's original frequency is taken without the virus. Once the virus is attached, the extra mass changes the frequency. Because of silicon's stability and ability to conduct lasers effectively, even tiny changes in mass can be detected through changes in oscillation frequency. The use of such a precise device allows for highly sensitive measurements, which are key in processes that require exact mass determination of microscopic particles.