Question

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 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.

Required:
a. 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.
b. In some data, the silicon sliver has a mass of 2.13×10^-16 g and a frequency of 2.04×10^15 Hz without the virus and 2.85×1014 with the virus. What is the mass of the virus in grams?

Answer 1

Answer:

a)   m_v = m_s (($\frac{w_o}{w}$)² - 1) ,  b)  m_v = 1.07 10⁻¹⁴ g

Explanation:

a) The angular velocity of a simple harmonic motion is

           w² = k / m

where k is the spring constant and m is the mass of the oscillator

let's apply this expression to our case,

silicon only

         w₉² = $\frac{K}{m_s}$

         k = w₀² m_s

silicon with virus

         w² = $\frac{k}{m_s + m_v}$

          k = w² (m_v + m_s)

in the two expressions the constant k is the same and q as the one property of the silicon bar, let us equal

           w₀²  m_s = w² (m_v + m_s)

           m_v = ($\frac{w_o}{w}$)²  m_s - m_s

           m_v = m_s (($\frac{w_o}{w}$)² - 1)

b) let's calculate

          m_v = 2.13 10⁻¹⁶ [($\frac{20.4}{2.85}$)² - 1)]

          m_v = 1.07 10⁻¹⁴ g