Question

In February 2004, scientists at 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 32.0 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.

Find the ratio of the frequency with the virus attached ( fS+V) to the frequency without the virus (fS) in terms of mV and mS, where mV is the mass of the virus and mS is the mass of the silicon sliver. Notice that it is not necessary to know or measure the force constant of the spring.

Express your answer in terms of the variables mV and mS

(fs+v)/fs = ?

Answer 1

The answer in
`(f_(S+V)/f_(S)) = sqrt((m_(S) + m_(V))/m_(S))`.

In a mass-spring system, the frequency of oscillation (f) is related to the mass (m) attached to the spring. The formula for the frequency (f) is given by:

`f=1/2\pi √(k/m)`

where k is the force constant of the spring. In this case, we are considering a silicon sliver (mass mS ) without the virus and then with the virus attached (mass mV​).

The ratio of the frequencies with the virus attached (fS+V​) to without the virus (fS​) can be expressed as:

`(f_(S+V)/f_(S)) = 1/2\pi √(k/m_s+m_v)/ \/1/2\pi √(k/m_s)`

Simplifying, the 2π terms cancel out, and the expression becomes:

`(f_(S+V)/f_(S)) = \sqrt{((m_(S) + m_(V))/m_(S))}`

This is the ratio of the frequencies with the virus attached to without the virus, and it's given in terms of the masses mS​ (mass of the silicon sliver) and mV (mass of the virus).