Chapter 14, Problem 14.22E

BIO 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 (f_S+V) to the frequency without the virus (f_S) is given by $f_{\text{S+V}}/f_{\text{S}}=1/\sqrt{1+\left(m_{\text{V}}/m_{\text{S}}\right)}$, where m_V is the mass of the virus and m_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 × 10⁻¹⁶g and a frequency of 2.00 × 10¹⁵ Hz without the virus and 2.87 × 10¹⁴ Hz with the virus. What is the mass of the virus, in grams and in femtograms?