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Chapter 14: Problem 22

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 \(30 \mathrm{nm}\) 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 \(f_{\mathrm{S}+\mathrm{V}} / f_{\mathrm{S}}=1 / \sqrt{1+\left(m_{\mathrm{V}} / m_{\mathrm{S}}\right)},\) 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 in femtograms?

Short Answer

Expert verified
The mass of the virus can be found from the solution steps and will depend on your previous calculation and conversion.

Step by step solution

01

Computing mass ratio

Starting with the known formula for frequency \(f = \frac{1}{2\pi} \sqrt{\frac{k}{m}}\), where k is the spring constant, m is the mass, we can substitute \(f_{\mathrm{S} + \mathrm{V}}\) and \(f_{\mathrm{S}}\) into the frequency formula and then take the ratio of the two.
02

Simplification

This yields \( \frac{f_{\mathrm{S} + \mathrm{V}}}{f_{\mathrm{S}}} = \frac{\frac{1}{2\pi} \sqrt{\frac{k}{m_{\mathrm{S}} + m_{\mathrm{V}}} }}{\frac{1}{2\pi} \sqrt{\frac{k}{m_{\mathrm{S}}} }}\). Simplifying this we get \(f_{\mathrm{S} + \mathrm{V}} /f_{\mathrm{S}} = \sqrt{\frac{m_{\mathrm{S}}}{m_{\mathrm{S}} + m_{\mathrm{V}}}} \). Rearranging terms and squaring both sides of the equation, we get \(f_{\mathrm{S}+\mathrm{V}}^2 / f_{\mathrm{S}}^2 = m_{\mathrm{S}}/(m_{\mathrm{S}} + m_{\mathrm{V}})\). This implies, \( m_{\mathrm{V}} = m_{\mathrm{S}} \left(\frac{1}{f_{\mathrm{S}+\mathrm{V}}^2 / f_{\mathrm{S}}^2} - 1\right)\).
03

Substituting known values

Taking the given values for \(m_{\mathrm{S}}\), \(f_{\mathrm{S}}\) and \(f_{\mathrm{S}+\mathrm{V}}\), we substitute them into the formula found in Step 2 to find \(m_{\mathrm{V}}\). This gives us \( m_{\mathrm{V}} = 2.10 \times 10^{-16} g \left(\frac{1}{\left(2.87 \times 10^{14} Hz \right)^2 / \left(2.00 \times 10^{15} Hz\right)^2} - 1\right)\).
04

Calculation and Conversion

Solving this yields the mass of the virus in grams. To convert it to femtograms (fg), we use the fact that 1g = \(10^{15}\) fg : \( m_{\mathrm{V}} = X g = Y fg \), where X and Y will depend on the previous calculation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mass on a Spring
The concept of a mass on a spring is a fundamental illustration of harmonic motion in physics. It is a situation where a mass is attached to the end of a spring, and when the mass is displaced from its equilibrium point, it exerts a force that tries to restore it to the original position. This results in the mass oscillating back and forth around the equilibrium, with the motion described by Hooke's law, which states that the force exerted by the spring is proportional to the displacement.
In the context of measuring virus mass, the silicon sliver acts like the mass on a spring system, and the virus acts as an additional mass that alters the oscillation of the system. The fundamentals behind this concept are crucial because understanding the relationship between mass and oscillation frequency allows scientists to measure the viral mass indirectly by observing changes in oscillation frequencies.

Importance of Equilibrium Position

When no external forces act on the mass-spring system, it comes to rest at the equilibrium position. This position is essential because it acts as a reference point for measuring oscillations. It is only when the mass is displaced from this position that the spring exerts a restoring force, which leads to an oscillation around the equilibrium.

Oscillation and Damping

Over time, most real-world oscillations experience damping due to external factors like air resistance, which causes the amplitude of the oscillation to decrease. However, in an ideal mass on a spring system, there is no damping, and the oscillations can continue indefinitely with a constant amplitude and frequency.
Oscillation Frequency
Oscillation frequency refers to the number of complete oscillations that a mass on a spring system undergoes per unit of time. It is a key concept in studying the dynamics of oscillatory systems and is measured in hertz (Hz). The frequency of an ideal undamped harmonic oscillator is dependent on two factors: the mass of the object and the spring constant of the spring to which it is attached.
In the exercise, the frequency changes when the virus attaches to the silicon sliver, because the system's mass increases, thereby affecting the frequency of oscillation. This change in frequency can be used to measure the mass of the virus using the ratio of the frequencies before and after the virus attachment.

Resonance

Resonance occurs when the frequency of an external force matches the natural oscillation frequency of the system, resulting in a significant increase in the amplitude of the oscillations. Resonance is not directly related to the exercise but is an important phenomenon to understand in the study of oscillatory systems.
Spring Constant
The spring constant, commonly denoted by the symbol k, is a measure of the stiffness of a spring. It indicates how much force is needed to extend or compress the spring by a certain displacement distance. The spring constant is central to Hooke's law, which expresses the restoring force of the spring as F = -kx, where x is the displacement from the equilibrium position.
In the exercise dealing with virus mass, understanding the spring constant is inherent to the model of a mass on a spring system. Although the exercise mentions that it is not necessary to know the spring constant to measure the mass of the virus, the spring constant still plays a critical role in determining the system's natural frequency.

Calculating Spring Constant

While not directly required for the exercise, calculating the spring constant involves measuring the force applied to the spring and the resulting displacement. Mathematically, it is expressed as k = F/x. This can also be determined experimentally by measuring the period of oscillation or the frequency of a mass-spring system and using the mass to calculate k.

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Most popular questions from this chapter

An object is moving in damped SHM, and the damping constant can be varied. If the angular frequency of the motion is \(\omega\) when the damping constant is zero, what is the angular frequency, expressed in terms of \(\omega\), when the damping constant is one-half the critical damping value?

The jerk is defined to be the time rate of change of the acceleration. (a) If the velocity of an object undergoing SHM is given by \(v_{x}=-\omega A \sin (\omega t),\) what is the equation for the \(x\) -component of the jerk as a function of time? (b) What is the value of \(x\) for the object when the \(x\) -component of the jerk has its largest positive value? (c) What is \(x\) when the \(x\) -component of the jerk is most negative? (d) When it is zero? (e) If \(v_{x}\) equals \(-0.040 \mathrm{~s}^{2}\) times the \(x\) -component of the jerk for all \(t,\) what is the period of the motion?

Consider the system of two blocks and a spring shown in Fig. \(\mathrm{P} 14.66 .\) The horizontal surface is friction less, but there is static friction between the two blocks. The spring has force constant \(k=150 \mathrm{~N} / \mathrm{m} .\) The masses of the two blocks are \(m=0.500 \mathrm{~kg}\) and \(M=4.00 \mathrm{~kg} .\) You set the blocks into motion by releasing block \(M\) with the spring stretched a distance \(d\) from equilibrium. You start with small values of \(d,\) and then repeat with successively larger values. For small values of \(d,\) the blocks move together in SHM. But for larger values of \(d\) the top block slips relative to the bottom block when the bottom block is released. (a) What is the period of the motion of the two blocks when \(d\) is small enough to have no slipping? (b) The largest value \(d\) can have and there be no slipping is \(d=8.8 \mathrm{~cm} .\) What is the coefficient of static friction \(\mu_{\mathrm{s}}\) between the surfaces of the two blocks?

A holiday ornament in the shape of a hollow sphere with mass \(M=0.015 \mathrm{~kg}\) and radius \(R=0.050 \mathrm{~m}\) is hung from a tree limb by a small loop of wire attached to the surface of the sphere. If the ornament is displaced a small distance and released, it swings back and forth as a physical pendulum with negligible friction. Calculate its period. (Hint: Use the parallel-axis theorem to find the moment of inertia of the sphere about the pivot at the tree limb.)

A uniform, solid metal disk of mass \(6.50 \mathrm{~kg}\) and diameter \(24.0 \mathrm{~cm}\) hangs in a horizontal plane, supported at its center by a vertical metal wire. You find that it requires a horizontal force of \(4.23 \mathrm{~N}\) tangent to the rim of the disk to turn it by \(3.34^{\circ},\) thus twisting the wire. You now remove this force and release the disk from rest. (a) What is the torsion constant for the metal wire? (b) What are the frequency and period of the torsional oscillations of the disk? (c) Write the equation of motion for \(\theta(t)\) for the disk.
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