Analogy between a capacitor and a spring. (a) How is adding charge to a capacitor like compressing a spring? (b) Why is it that the inverse of C

3. Plug the guess back into the system of DEQs to see if it is actually a solution, and to determine whether there are any restrictions on the parameters that appear

1.12 Analogy between a capacitor and a spring. (a) How is adding charge to a capacitor like compressing a spring? (b) Why is it that the inverse of C

isomorphic tok, rather than justCitself ?

1.13 In research-level theoretical physics, it is almost never possible to get an exact solution because of the complexity of the problems being considered.

Therefore, it is essential to make appropriate approximations, so as to get physical insight. The Taylor series is central to many of these approximations.

You have already seen two of the three most common applications of the Taylor series: sinθ ∼=θ(for small θ) and cosθ ∼= 1− θ²

2 (for smallθ).

In this problem, you will demonstrate the third of the three most common applications. Show that (1+x)^n∼₌1+nxforx1. (Note that this works whethernis positive or negative, integer or fractional.)

1.14 The potential energy for a particular object isU(x)= −Lcosβx, whereL andβare both>0. (This potential energy function is important in the study of superconductivity.)

(a) Make a sketch of this potential energy fromx= −2π

β tox= +2π β . Indicate the scale on the vertical axis.

(b) The object has massmand total energyE_TOT= −L+G, where 0<GL. (The symbol “” means “much less than.”) Add a dashed line to your sketch indicating this total energy.

(c) At t = 0, the object is at x = 0. Show that its motion can be approximated by a simple harmonic oscillation, and find the approximate frequency of oscillation. Hint: recall that the Taylor series expansion for cosθ is cosθ =1−θ²

2_! ⁺ θ⁴

4_! ^{− · · ·},so that forθ 1, cosθ∼=1₋^θ

2

2!.

1.15 LetC₁andC₂be complex numbers. Show thatC₁ C₂

= C₁

C₂.Reminder:

C₁means “magnitude of C₁”

1.16 For each of the following, express the quantity shown either in the forma₊ib(i.e., Cartesian representation)or in the formAe^iα (i.e., polar representation), whichever you find easier for each part of the problem.

In case you might be confused by the way I’ve written things: “i 4.5” means

“i times 4.5.”

(a) (3.2 + i 6.7) + (5.6 – i 4.5) (b) 6.1 e^{i 1.2}+ 1.2 e^{i 1.7}

(c) (3.2 + i 6.7)(5.6 – i 4.5) (d) (6.1 e^{i 1.2})(1.2 e^{i 1.7})

(e) What point is this problem trying to get across?

1.17 Letz₁=8e^iπ/6andz₂=√ 2e^i3π/4.

(a) Representz₁andz₂in the complex plane.

(b) Find the real and imaginary parts ofz₁andz₂.

Express each of the following in the formAe^iϕ, whereAandϕare real:

(c) z₁+z₂ (d) z³₁z²₂

1.18 A mass m= 10 kg is oscillating on a spring with k= 10 N/m with little damping. The displacement of the mass can be described by

x(t)₌ReCe^iωt,whereC₌(1 – i) cm.

(a) What is the value ofω?

(b) What is the amplitude of the motion?

(c) The solution can also be expressed in the formx(t)=Acos(ωt+ϕ).

What is the value ofϕ?

(d) Describe the initial conditions of the motion, that is, specify the position and velocity att= 0. As for all numbers in physics (except dimensionless quantities) besureto include units!

(e) Sketch two graphs, one of position versus time and the other of velocity versus time. Be sure to label both axes of each graph quantitatively.

(f) What is the energy of the oscillator?

1.19 For an RC low-pass filter (figure 1.10.2b), show thatV_OUT V_IN

drops by a factor of 10 for each factor of 10 increase inω, forωω_LO.

1.20 Using methods similar to those leading to equation (1.10.2), show that the complex impedances of two circuit elements in parallel combine in the way specified by equation (1.10.3).

1.21 A voltageV(t)=V₀cosωtis applied across a capacitor with capacitance C. (a)Without using a symbolic algebra program or graphing calculator, make a sketch with curves for both V(t) and I(t), showing two full periods of oscillation. Label your sketch quantitatively. (b)For a capacitor, does the current “lead” the voltage (meaning that the current reaches a peak before the voltage does), or does the voltage lead the current?

(c)Using simple ideas about charging of the capacitor and the connection between voltage and charge, explain why your answer to part (b) makes sense.

1.22 A voltageV(t)₌V₀cosωtis applied across an inductor with inductanceL.

(a)Without using a symbolic algebra program or graphing calculator, make a sketch with curves for bothV(t) andI(t), showing two full periods of oscillation. Label your sketch quantitatively.(b)For an inductor, does the current “lead” the voltage (meaning that the current reaches a peak before the voltage does), or does the voltage lead the current?(c)Using elements from the isomorphism between an LC oscillator and a mass/spring system, explain why your answer to part (b) makes sense.

1.23 A voltageV(t)=V₀cosωtis applied to the input of an RC low-pass filter (figure 1.10.2b). At the output, one observes a voltageV_OUTcos(ωt₊ϕ).

(a)Findϕ as a function ofω,R, andC. (b)Using whatever software you like, make two graphs of -ϕas a function ofω/ω_LO, ranging fromω/ω_LO= 0.01 to 100, one plot with linear axes and one with logarithmic axes.

1.24 The RC high-pass filter. For the circuit shown in figure 1.P.2, a sinusoidal input voltage is applied relative to ground, and the resulting output voltage is measured relative to ground. If a sinusoidal voltageV_in=V_icosωtis applied to the input, one observes a sinusoidal voltageV_{out =}V_ocos(ωt₊ϕ) at the output (relative to ground).(a) Show that amplitude of output voltage

amplitude of input voltage ⁼ V_o

V_i ⁼ 1

1₊ ω_HI

ω

2, where ω_HI ≡ 1

RC. (b) Show that the phase shift of the output relative to the input is ϕ = tan⁻¹

ω_HI ω

. (c) Given that ϕ is positive, does the output “lag” the input (i.e., do the peaks in the output voltage occur after the peaks in the input), or does the output “lead”

the input (i.e., do the peaks in the output occur before the peaks in the input)?

1.25 More on the RC high-pass filter.Read all of problem 1.24 before beginning this problem. (a) Using a suitable computer program, make two plots of

amplitude of output voltage amplitude of input voltage ⁼

V_o

V_i versus ^ω

ω_HI. In your first plot, use linear scales for both axes. In your second plot, use logarithmic scales for both axes. For both plots, let ^ω

ω_HIvary from 0.01 to 100. (b)Make two plots ofϕ versus ^ω

ω_HI. In your first plot, use linear scales for both axes. In your second plot, use a linear scale for theϕaxis and a logarithmic scale for the ^ω

ω_HIaxis.

Figure 1.P.2 The RC high-pass filter.

For both plots, let ^ω

ω_HIvary from 0.01 to 100. (c)Forωω_HI, show that amplitude of output voltage

amplitude of input voltage ⁼ V_o

V_i drops by a factor of 10 for each factor of 10 decrease inω.

1.26 More on the RC low-pass filter. (a)Consider the RC low-pass filter shown in figure 1.10.2b. If a voltage (relative to ground)V_in=V_icosωtis applied to the input, one observes a sinusoidal voltageV_out =V_ocos (ωt+ϕ) at the output (relative to ground). Show that the phase shift of the output voltage relative to the input isϕ =tan⁻¹

− ω ω_LO

, whereω_LO≡ 1

RC. (b)Given thatϕ is negative, does the output “lag” the input (i.e., do the peaks in the output voltage occur after the peaks in the input), or does the output

“lead” the input (i.e., do the peaks in the output occur before the peaks in the input)?

1.27 The RL low-pass filter.One can use resistors and capacitors to build either low-pass filters (see section 1.10) or high-pass filters (see problems 1.24 and 1.25). It is also possible instead to use resistors and inductors for these tasks; we’ll explore the low-pass version in this problem. In practical circuits, RC filters are much more common than RL filters, partly because it is easier to build an essentially ideal capacitor than to build an ideal inductor. (There is always resistance in the wires used to wind an actual inductor, and there is always capacitance between the windings.) Also, inductors are generally bulkier and more expensive than the capacitors that could be used to build comparable filters. However, one does see RL filters in some high frequency applications. Consider the circuit shown in figure 1.10.4. If a voltage (relative to ground)V_in=V_icosωtis applied to the input, one observes a sinusoidal voltageV_{out =}V_ocos(ωt₊ϕ) at the output (relative to ground). (a) Show that amplitude of output voltage

amplitude of input voltage ⁼ V_o

V_i ⁼

1 1+

ω ω_LO

2, whereω_LO ≡ R

L. (b)Show that the phase shift of

the output relative to the input isϕ =tan⁻¹

− ω ω_LO

. (Note that, except for the definition of ω_LO, these expressions are the same as for the RC low-pass filter.)

1.28 An electron in an atom can be excited from its original “ground state” to a well-defined and reproducible higher energy metastable “excited state.”

The electron can then “fall” back down to the ground state, emitting a photon in the process which has energy equal to the difference between the ground state and the excited state. (As it turns out, this energy is characteristic of the type of atom, and so analysis of such photons can be used for determining the elements which are present in a sample.) Many of these excited states have only a very short lifetime. For such a state,

if the experiment is performed carefully, one can determine that the photons emitted from a large sample of material do not all have exactly the same energy, but instead there is a small spread. If a particular excited state has a lifetime of 10⁻⁷s, about how big a spread in photon energy would you expect?Hint: Use the energy-time uncertainty relation; this is meant to be a very easy problem.

2 Examples of Simple Harmonic Motion

And I saw the cantilever jutting through the mist, resplendent in the light of dawn, oscillating jauntily like a promise of joy.

–Marian McKenzie

2.1 Requirements for harmonic oscillation

In this chapter, we will explore several examples of the remarkable variety of systems that show the harmonic oscillations described in chapter 1. There are two basic requirements for a system to oscillate: (1) If the system is disturbed from equilibrium, there must be something (such as a force) that tends to bring it back toward equilibrium.

For the oscillations to be of the sinusoidal form described in chapter 1, this restoring drive must be proportional to the displacement from equilibrium, for example, the spring force F is proportional to x: F = −kx. (2) As the system moves toward equilibrium, there must be something (such as inertia) which tends to make the system overshoot the equilibrium point.

Saying the same thing mathematically, if the system is described by a differential equation of the form

F = −kx⇒ md²x dt² !

inertial term:

satisfies condition 2

= _!−kx

restoring term:

satisfies condition 1

, (2.1.1)

then it will exhibit harmonic oscillations.

Sometimes, it is easier to consider the energy of a system. If the system can be described by an equation of the form

1 2m

dx dt

2

!

kinetic energy

+ ¹₂kx² !

potential energy

=constant, (2.1.2)

then it will exhibit harmonic oscillation.

39

In other words, if we can show that a system obeys an equation either of the form (2.1.1) or of the form (2.1.2), then we can immediately conclude that

x=Acosω₀t+ϕ

, whereω₀= k

m^. (2.1.3)

2.2 Pendulums

Professor Roger Newton, author of the book Galileo’s Pendulum, recounts this wonderful legend about Galileo, the world’s first experimental physicist, and a revelation that occurred during a church service in 1581:

He was seventeen and bored listening to the Mass being celebrated in the cathedral of Pisa. Looking for some object to arrest his attention, the young medical student began to focus on a chandelier high above his head, hanging from a long, thin chain, swinging gently to and fro in the spring breeze. How long does it take for the oscillations to repeat themselves, he wondered, timing them with his pulse. To his astonishment, he found that the lamp took as many pulse beats to complete a swing when hardly moving at all as when the wind made it sway more widely.¹

This description of one of the first quantitative observations of experimental physics shows the historic importance of pendulums in physics. We will see in chapter 4 that pendulums provide an excellent illustration of chaos theory. Pendulums are common in everyday life, from a baby’s swing to a grandfather clock, from a fair ride to a wrecking ball.

We recognize Galileo’s observation that the period is independent of the amplitude, as a characteristic of simple harmonic motion. In the case of the pendulum, although there is an obvious mass, there is no obvious spring. Yet, since it does have a position of stable equilibrium, we should be able to model the potential energy near this position as a parabola,and so we should be able to find an “effective spring constant” that arises from the combination of gravity and the tension in the string.

Let’s consider an arbitrary rigid object of massmthat can rotate in thex–yplane about a pivot P, as shown in figure 2.2.1. We’ll show that, for small amplitudes of swing, the energy takes the form (2.1.2). In the figure, we displace the pendulum by an angleθfrom equilibrium. The potential energy is determined by the position of the center of mass (marked CM in the figure):U₌mgy, whereyis the height of the CM above its equilibrium position.

Your turn:Show that, ifθis small, then

U∼= ¹2 mg_CMθ². (2.2.1)

1. Roger G. Newton,Galileo’s Pendulum: From the Rhythm of Time to the Making of Matter, Harvard University Press, Cambridge, MA, 2004, p. 1.

Figure 2.2.1A pendulum of arbitrary form.

Hint: Use the Taylor expansion for cosθ, which you derived in section 1.6:

cosθ=1−θ² 2! +θ⁴

4! − · · · (1.6.5) ( Ifθis small, this means that cosθ∼=1− θ²

2!.)

The motion is a pure rotation about P, so the kinetic energy isK ₌ ¹₂Iω², where I is the moment of inertia for rotations about P andω = ˙θ is the angular velocity.

Combining this with your result (2.2.1) gives the total energy:

E=U+K= ¹₂mg_CMθ²+¹₂Iθ^˙2. (2.2.2) Since the total energy is constant, this has exactly the same form as

(2.1.2): ¹₂kx²+¹₂mx˙²= constant,

so that we immediately know that the pendulum displays simple harmonic motion, that is, that

θ=θ₀cosω₀t+ϕ

,whereω₀=

mg_CM

I ^. (2.2.3)

Motion of a pendulum (mass need not be concentrated at a point).

I=moment of inertia about pivot,_CM=length from pivot to CM

Core example: the simple pendulum. The simplest example of a pendulum is a compact mass (the “pendulum bob”) at the end of a thin rod of length; we assume the mass of the rod is negligible compared to that of the bob. In this case, the moment of inertia about the pivot point isI=m². Plugging this into equation (2.2.3) gives

ωsimple pendulum

= g

. (2.2.4)

For more complicated objects, one often uses the parallel axis theorem, which you may have seen proved in an introductory physics book:

I=I_CM+mh², (2.2.5)

The parallel axis theorem

whereI_CMis the moment of inertia for rotations about the center of mass andhis the distance from the pivot point P to the center of mass. By breaking a complicated object up into smaller symmetrical objects and applying the parallel axis theorem, one can computeIof the complicated object.

Although the harmonic motion of the pendulum is most easily seen by considering the time dependence ofθ (as we have done earlier), we can also show that there is a harmonic variation in the horizontal positionx. For a simple pendulumI =m², so that equation (2.2.2) becomes

E= ¹₂mgθ²+¹₂m²θ˙²= ¹₂mg

²θ²+¹₂m²θ˙²=¹₂mg

(θ)²+¹₂m

"

d dt(θ)

#2

.

As we can see from the figure, in the limit of small displacements, the arc lengthθ∼=x, so that

E^∼₌¹₂ mg

x²₊¹₂mx_˙².

Since this has the same form as the energy of a mass/spring system,E₌ ¹₂kx²₊¹₂mx_˙², we see that the effective spring constant for the pendulum is k_{pendulum =} mg

. This means that the net restoring force, which is created by a combination of gravity and the string tension, is

F_{pendulum= −}k_pendulumx_{= −}mg

x. (2.2.6)

“Pendulum force” resulting from the combination of gravity and tension for a simple pendulum.

We will use this result again in chapter 5.

2.3 Elastic deformations and Young’s modulus

All materials are at least a little stretchy, although the amount an object can be stretched before breaking is often too small to see with the naked eye. This stretchiness, and the vibrations that occur when an object is stretched or twisted and then released, determine the engineering limits of building materials, the performance limits of automotive components, and the behavior of a new class of devices known as Micro ElectroMechanical Systems (MEMS).²In the remainder of this chapter, we’ll explore

2. These devices, which combine mechanical motion with electrical actuation or sensing, are fabricated using techniques of photolithography, electron beam lithography, and various types of

Figure 2.3.1 Top: a relaxed spring at its equilibrium length. Bottom: When the spring is stretched, the left and right halves stretch by equal amounts.

various types of elastic (i.e., reversible) deformations, and their importance in science and everyday life.

The simplest way to deform an object is to stretch or compress it. Consider a long object of uniform cross-section, such as a beam, which is anchored at the left end. If a force is applied to the right end, one always observes that the beam stretches by an amount proportional to the force. This comes as no surprise – before the force is applied, the beam is in equilibrium, and by the arguments in chapter 1 any displacement from equilibrium is countered by a forceF_{spring= −}kx. Therefore, to produce a displacement x, we must applyF_applied= −F_spring=kx.

The spring constantk depends on the material from which the beam is made;

diamond is stiffer than rubber. However,kalso depends on the cross-sectional area and length of the beam. We wish to divide out these geometric dependencies to get a parameter that describes the springiness or stiffness of the material itself. First, we consider the dependence on length. How doeskchange if the beam is cut to half its length? Since we’re modeling the beam as a spring, this is equivalent to asking what happens to the spring constant of a spring when it is cut in half.

Imagine a spring of equilibrium lengthwhich is attached to a wall on the left side.

A dog pulls on the right end, stretching it by an amount, as shown in figure 2.3.1.

The force exerted by the spring on the dog is F_{by spring}= −k

⇔k= −F_{by spring}

= − F_{by spring}

extension^. (2.3.1)

anisotropic (meaning directionally dependent) etching. The sizes of the devices range from the diameter of a human hair down to the molecular regime, allowing extremely fast response times and, for devices designed to detect trace chemicals, extraordinary sensitivity. We will discuss MEMS devices in sections 2.6 and 3.4. You can learn more about MEMS in problem 2.14, and in the website for this text, under the entry for this section.

The dog must be exerting an equal and opposite forcekon the right end of the spring. Since the right half of the spring doesn’t accelerate, we know there must be another force to balance this; this force is exerted by the left half on the right half

Fby left half

= −k.

The extension of the left half is

2 . Therefore, by analogy with equation (2.3.1), the spring constant of the left half is

k_left

half

= − Fby left

half

extension ^{= −}

(−k) (/2) ⁼2k.

Therefore, when a spring is cut in half, the spring constant gets doubled. (Another way to see this: for the same extension, the coils of a short spring are distorted more than the coils of a long spring, therefore the shorter spring exerts a bigger force.)

Of course, this also means that, if the length of the spring (or in our case the length of the beam) is doubled, the spring constant is halved. Thus,k∝ 1

. (The symbol∝ means “proportional to.”)

Your turn (answer³at bottom of page): Explain whyk is proportional to the cross- sectional areaAof the beam.

Putting these results together, we can writek∝ A or k=EA

, (2.3.2)

where the proportionality constant E is called “Young’s Modulus,”⁴ and is the equilibrium lengthof the beam.

We can also write

F_spring= −kx= −EA

x⇔F_spring

A ^{= −}Ex .

The force appliedtothe beam,F_applied, is equal and opposite to the forceF_springapplied bythe beam, so that

F_applied A ⁼Ex

. (2.3.3)

3. Answer: We can divide the beam intoNsmaller beams running in parallel along the length. The force from a single one of these would beF_small= −k_smallx, wherexis the amount by which the beam is stretched. The force from the entire beam is the total of the forces from the small beams:F_TOT=NF_small= −

Nk_small

x, so that the total spring constant isk=Nk_small. If the cross-sectional areaAis doubled, thenNdoubles, sok∝A.

4. This is named after Thomas Young, who is most famous for his 1801 two-slit experiment, which established the wave nature of light. Young was also a physician, and figured out how the eye focuses on objects at different distances. He was familiar with twelve languages by the age of fourteen, and later was involved in the translation of the Rosetta stone.