In February 2001, scientists at Purdue University used a highly sensitive techni
ID: 1425100 • Letter: I
Question
In February 2001, scientists at Purdue University used a highly sensitive technique to measure the mass of a vaccine virus (the kind used in smallpox vaccine). The procedure involved measuring the frequency of oscillation of a tiny sliver of silicon (just 31.0 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 (integral_s+v) to the frequency without the virus (integral_s) in terms of m_v and m_s, 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 known or measure the force constant of the spring. In some data, the silicon sliver has a mass of 2.13 time sign 10^-16 g and a frequency of 2.04 time sign 10^15 Hz without the virus and 2.86 time sign 10^14 Hz with the virus. What is the mass of the virus in grams? In some data, the silicon sliver has a mass of 2.13 time sign 10^-16 g and a frequency of 2.04 time sign 10^15 Hz without the virus and 2.86 time sign 10^14 Hz with the virus. What is the mass of the virus in grams? What is the mass of the virus in femtograms?Explanation / Answer
A period of simple harmonic motion done by mass-spring system, is given by
T = 2 (m/k) and its frequency f = 1/T by f = 1/2 (k/m).
The force constant of the spring is the same in both cases, so we write two equations:
f(s+v) = 1/2 (k / (ms + mv) ) and
f(s) = 1/2 (k / ms).
The ratio of both frequencies is
f(s+v) / f(s) = 1/2 (k / (ms + mv) ) / 1/2 (k / ms).
The force constant k and the 2 factors are cancelled out, and we obtain:
f(s+v) / f(s) = (ms / (ms + mv)). =
To get the mass of the virus, we solve the above equation for mv.
We first square both sides of the equation to eliminate the square root:
[ f(s+v) / f(s) ]² = ms / (ms + mv).
Now multiply both sides by
(ms + mv): [ f(s+v) / f(s) ]² (ms + mv) = ms ==> [ f(s+v) / f(s) ]² mv = ms – [ f(s+v) / f(s) ]² ms
==> mv = ms – [ f(s+v) / f(s) ]² ms / [ f(s+v) / f(s) ]²
= 2.13×10^-16 g – (2.86×10^14 Hz / 2.04×10^15 Hz)² x 2.13×10^-16 g / (2.86×10^14 Hz / 2.04×10^15 Hz)²
= 1.07 x 10^–14 g = 10.7 fg
(femtograms, 1 fg = 10^–15 g).