Questions and Answers of Modern Classical Physics

In any high-precision Foucault pendulum, it is important that the pendular restoring force be isotropic, since anisotropy will make the swinging period different in different planes and thereby cause
From Eq. (11.18) derive expression (11.30) for the elastostatic force density inside an elastic body. T = -K0g - 2μΣ. (11.18)
(a) Verify Eqs. (11.46) for the sag in a horizontal beam clamped at one end and allowed to hang freely at the other end.(b) Now consider a similar beam with constant cross section and loaded with
(a) Consider a W = ∇ξ that is pure expansion: Using Eq. (11.3) show that, in the vicinity of a chosen point, the displacement vector is Draw this displacement vector field.(b) Similarly, draw
For an anisotropic, elastic medium with elastic energy densityintegrate this energy density over a 3-dimensional region V (not necessarily small) to get the total elastic energy E. Now consider a
Using Eqs. (1), (2), and (3) of Box 11.2, show thatis equal to Wij.Data from Eqs. (1), (2), and (3) of Box 11.2. ©g;; + Σ;; + Rij gij
Beginning with[text following Eq. (11.23)], derive Eq. (11.24) for the elastic energy density inside a body. U =-{{Tij Ši;j
Explain why all animals, from fleas to humans to elephants, can jump to roughly the same height. The field of science that deals with topics like this is called allometry (Ex. 11.18).Data from
(a) What is the maximum size of a nonspherical asteroid?(b) What length of steel wire can hang vertically without breaking? What length of carbon nanotube? What are the prospects for creating a
Suppose a light beam is split in two by a beam splitter. One beam is reflected off an ordinary mirror and the other off a phase-conjugating mirror. The beams are then recombined at the beam splitter.
Describe the creation and annihilation of photons that underlies a phase conjugating mirror’s four-wave mixing. Specifically, how many photons of each wave are created or annihilated?
Suppose the thickness of the nonlinear medium of the text’s four-wave mixing analysis is L = π/(2κ), so the denominators in Eqs. (10.67) are zero. Explain the physical nature of the resulting
Use the method of moments (Sec. 11.5) to derive the 2-dimensional shape equation (11.63a) for the stress-induced deformation of a thin plate, and expression (11.63b) for the 2-dimensional flexural
Show how to construct a paraboloidal mirror of radius R and focal length f by stress polishing.(a) Adopt a strategy of polishing the stressed mirror into a segment of a sphere with radius of
Consider a slender wire of rectangular cross section with horizontal thickness h and vertical thickness w that is resting on a horizontal surface, so gravity is unimportant. Let the wire be bent in
Derive Eq. (11.55) relating the angle θo = (dη/dx)x = 0 = kηo = πηo/ℓ to the applied force F when the card has an n = 1, arched shape.(a) Derive the first integral of the elastica
Explore numerically the free energy (11.57) of a bent beam with a compressive force F and lateral force Flat. Examine how the extrema (equilibrium states) evolve as F and Flat change, and deduce the
Allometry is the study of biological scaling laws that relate various features of an animal to its size or mass. One example concerns the ratio of the width to the length of leg bones. Explain why
A DNA molecule consists of two long strands wound around each other as a helix, forming a cylinder with radius a ≈ 1nm. In this exercise, we explore three ways of measuring the molecule’s
Derive Eqs. (11.76) and (11.77) for the divergence of the vector field ξ in cylindrical and spherical coordinates using the connection coefficients (11.70) and (11.71).In Equations
(a) By drawing pictures analogous to Fig. 11.15, show that(b) From these relations and antisymmetry on the first two indices [Eq. (11.69)], deduce the connection coefficients (11.71).Equation 11.69
Derive Eqs. (11.92) for the cylindrical components of the internal elastostatic force per unit volumein a cylindrically symmetric situation.Eqs. (11.92) f = (K+u)V(V. §) + μV ²
A torsion pendulum is a very useful tool for testing the equivalence principle (Sec. 25.2), for seeking evidence for hypothetical fifth (not to mention sixth!) forces, and for searching for
Fill in the details of the text’s analysis of the deformation of a pipe carrying a high pressure fluid, and the wall thickness required to protect the pipe against fracture. (See Fig. 11.16.)Fig.
By a computation analogous to Eq. (11.72), derive Eq. (11.78) for the components of the gradient of a second-rank tensor in any orthonormal basis. Vk (Šjej) = (Vkšj)ej + šj (Vkej) = j,kej +
Suppose that a stress Tzjapplied (xo) is applied on the face z = 0 of a half-infinite elastic body (one that fills the region z >0). Then by virtue of the linearity of the elastostatics equation f
Modify the wave equation (12.4b) to include the effect of gravity. Assume that the medium is homogeneous and the gravitational field is constant. By comparing the orders of magnitude of the terms in
Just as in electromagnetic theory, it is sometimes useful to write the displacement ξ in terms of scalar and vector potentials:(The vector potential A is, as usual, only defined up to a gauge
Derive the energy-density, energy-flux, and lagrangian properties of elastodynamic waves given in Sec. 12.2.5. Specifically, do the following.(a) For ease of calculation (and for greater generality),
Verify Eqs. (12.35) and (12.37). Sketch the dispersion-induced evolution of a Gaussian wave packet as it propagates along a stretched beam. w² = C²k² =C²² (1+^). (12.35)
Consider a beam of length ℓ, whose weight is negligible in the elasticity equations, supported freely at both ends (so the slope of the beam is unconstrained at the ends). Show that the frequencies
Show that the sound speeds for the following types of elastic waves in an isotropic material are in the ratiosThe elastic waves are (i) Longitudinal waves along a rod, (ii) Longitudinal waves along
Derive the junction condition [Tjz] = 0 at a horizontal discontinuity between two media by the same method as one uses in electrodynamics to show that the normal component of the magnetic field must
Using the boundary conditions (12.44), show that at a surface of discontinuity inside Earth, SV and P waves mix, but SH waves do not mix with the other waves. [j] = [Tj₂] = 0, (12.44)
The magnitude M of an earthquake, on modern variants of the Richter scale, is a quantitative measure of the strength of the seismic waves it creates. The earthquake’s seismic-wave energy release
Consider a longitudinal elastic wave incident normally on the boundary between two media, labeled 1 and 2. By matching the displacement and the normal component of stress at the boundary, show that
Buckling plays a role in many natural and human-caused phenomena. Explore the following examples.(a) Mountain building. When two continental plates are in (very slow) collision, the compressional
The 4-acceleration of a particle or other object is defined by a(vector) ≡ du(vector)/dτ, where u(vector) is its 4-velocity and τ is proper time along its world line. Show that, if an observer
A galaxy such as our Milky Way contains ∼1012 stars—easily enough to permit a kinetic-theory description of their distribution; each star contains so many atoms (∼1056) that the masses of the
Consider a collection of identical test particles with rest mass m ≠ 0 that diffuse through a collection of thermalized scattering centers. (The test particles might be molecules of one species,
The universe is filled with cosmic microwave radiation left over from the big bang. At each event in spacetime the microwave radiation has a mean rest frame. As seen in that mean rest frame the
By performing a 3 + 1 split on the geometric version of Maxwell’s equations (2.48), derive the elementary, frame-dependent versionData from Equation 2.48 V.E=4л ре V.B = 0, V x B
Consider a fluid with 4-velocity u(vector) and rest-mass density ρo as measured in the fluid’s rest frame.(a) From the physical meanings of u(vector), ρo, and the rest-mass-flux 4-vector
Use the 2-dimensional spacetime diagrams of Fig. 3.4 to show that are frame independent [Eqs. (3.7a) and (3.7b)].Fig 3.4 EdVx and dVp/E
Consider a collection of thermalized, classical particles with nonzero rest mass, so they have the Boltzmann distribution. Assume that the temperature is low enough (kBT ≪ mc2) that they are
Consider a nonrelativistic fluid that, in the neighborhood of the origin, has fluid velocitywith σij symmetric and trace-free. As we shall see in Sec. 13.7.1, this represents a purely shearing flow,
Show that for thermalized, classical relativistic particles the probability distribution for the speed isWhere K2 is the modified Bessel function of the second kind and order 2. This is sometimes
(a) Cygnus X-1 is a source of X-rays that has been studied extensively by astronomers. The observations (X-ray, optical, and radio) show that it is a distance r ∼ 6,000 light-years from Earth. It
Consider a nonrelativistically degenerate electron gas at finite but small temperature.(a) Show that the inequalities kBT ≪ μe ≪ me are equivalent to the words “nonrelativistically
(a) Consider a collection of photons with a distribution function N that, in the mean rest frame of the photons, is isotropic. Show, using Eqs. (3.49b) and (3.49c), that this photon gas obeys the
A simplified version of a commercial nuclear reactor involves fissile material such as enriched uranium12 and a moderator such as graphite, both of which will be assumed in this exercise. Slow
(a) Show that the following is a solution to the diffusion equation (3.71) for particles in a homogeneous infinite medium:so N is the total number of particles. Note that this is a Gaussian
Consider a mode S of a fermionic or bosonic field. Suppose that an ensemble of identical such modes is in statistical equilibrium with a heat and particle bath and thus is grand canonically
Consider a collection of identical, classical (i.e., with η ≪ 1) particles with a distribution function N that is thermalized at a temperature T such that kBT ≪ mc2 (nonrelativistic
Derive the equations of state (3.52) for an electron-degenerate hydrogen gas. EF = μ = m₂ cosh(t/4), PF= |C –mẻ=mesinh(t/4). (3.52a)
Canonical transformations are treated in advanced textbooks on mechanics, such as Goldstein, Poole, and Safko or, more concisely, Landau and Lifshitz (1976). This exercise gives a brief introduction.
Derive Eq. (3.43) for the electron pressure in a nonrelativistic, electron-degenerate hydrogen gas. 5/3 2/3 3 m₂c² P. = 2/10 (²) ²0 m² (™. P Pe 23 mp/23) (3.43)
Use the law of energy conservation to show that, when heat diffuses through a homogeneous medium whose pressure is being kept fixed, the evolution of the temperature perturbation δT ≡ T −
Consider fully thermalized electromagnetic radiation at temperature T , for which the mean occupation number has the standard Planck (blackbody) form η = 1/(ex − 1) with x = hν/(kBT).(a) Show
Consider a collection of freely moving, noncolliding particles that satisfy the collisionless Boltzmann equation (a) Show that this equation guarantees that the Newtonian particle conservation law
Consider a universe (not ours!) in which spacetime is flat and infinite in size and is populated throughout by stars that cluster into galaxies like our own and our neighbors, with interstellar and
Make rough estimates of the entropy of the following systems, assuming they are in statistical equilibrium.(a) An electron in a hydrogen atom at room temperature.(b) A glass of wine.(c) The Pacific
Consider an ensemble of classical systems with each system made up of a large number of statistically independent subsystems, so ρ = ∏a ρa. Show that the entropy of the full ensemble is equal to
Following the epoch of primordial element formation (Ex. 4.10), the universe continued to expand and cool. Eventually when the temperature of the photons was ∼3,000 K, the free electrons and
Consider two Cartesian coordinate systems rotated with respect to each other in the x-y plane as shown in Fig. 1.4.Fig. 1.4.(a) Show that the rotation matrix that takes the barred basis vectors to
Derive, or at least give a plausibility argument for, Landauer’s theorem.
Consider a microcanonical ensemble of closed cubical cells with volume V . Let each cell contain precisely N particles of a classical, nonrelativistic, perfect gas and contain a nonrelativistic total
Consider a classical, nonrelativistic gas whose particles do not interact and have no excited internal degrees of freedom (a perfect gas—not to be confused with perfect fluid). Let the gas be
(a) Consider two identical chambers, each with volume V, separated by an impermeable membrane. Into one chamber put energy E and N atoms of helium, and into the other, energy E and N atoms of xenon,
Analyze the behavior of the atoms’ total energy near the onset of condensation, in the limit of arbitrarily large N (i.e., keeping only the leading order in our 1/N1/3 expansion and approximating
Show that in the Bose-Einstein condensate discussed in the text, the momentum distribution for the ground-state-mode atoms is Gaussian with rms momentumand that for the classical cloud it is Gaussian
Analyze Bose-Einstein condensation in a cubical box with edge lengths L [i.e., for a potential V (x, y, z) that is zero inside the box and infinite outside it]. In particular, using the analog of the
Derive Eq. (4.70) for the average number of bits per symbol in a long message constructed from N distinct symbols, where the frequency of occurrence of symbol n is pn. N 1 = LΣ-Pn log₂
Consider messages of length L >> 2 constructed from just two symbols (N = 2),which occur with frequencies p and (1− p). Plot the average information per symbol I̅ (p) in such messages, as a
Two dice are thrown randomly, and the sum of the dots showing on the upper faces is computed. This sum (an integer n in the range 2 ≤ n ≤ 12) constitutes a symbol, and the sequence of results of
Energy-Momentum Conservation for a Perfect Fluid(a) Derive the frame-independent expression (2.74b) for the perfect fluid stress energy tensor from its rest-frame components (2.74a).(b) Explain why
Consider, as in Ex. 2.9, an observer with 4-velocity U(vector) who measures the properties of a particle with 4-momentum p(vector).(a) Show that the Euclidean metric of the observer’s 3-space, when
An observer with 4-velocity U(vector) measures the properties of a particle with 4-momentum p(vector). The energy she measures is E = −p(vector) · U(vector) [Eq. (2.29)].(a) Show that the
Suppose that some medium has a rest frame(unprimed frame) in which its energy flux and momentum density vanish, T0j = Tj0 = 0. Suppose that the medium moves in the x direction with speed very small
(a) Express the electromagnetic field tensor as an anti-symmetrized gradient of a 4-vector potential: in slot-naming index notationShow that, whatever may be the 4-vector potential A(vector, the
(a) From Eqs. (2.75) and (2.45) compute the components of the electromagnetic stress-energy tensor in an inertial reference frame (in Gaussian units). Your answer should be the expressions given in
(a) Show that, if nj is a 3-dimensional unit vector and β and γ are defined as in Eq. (2.37b), then the following is a Lorentz transformation [i.e., it satisfies Eq. (2.35b)]:Show, further, that
(a) Convert the following expressions and equations into geometric, index-free notation(b) Convert T (___, S(R(C(vector), ___), ___), ___) into slot-naming index notation. Aa Bysi AaBys; SBY = SYBA B
Show that the matrices (2.37a), with β and γ satisfying Eq. (2.37b), are the inverses of each other, and that they obey the condition (2.35b) for a Lorentz transformation. [2] - ν βγ 0 0 βγ ο
(a) Simplify the following expression so the metric does not appear in it:(b) The quantity gαβgαβ is a scalar since it has no free indices. What is its numerical value?(c) What is wrong with the
Use spacetime diagrams to prove the following:(a) Two events that are simultaneous in one inertial frame are not necessarily simultaneous in another. More specifically, if frame F̅ moves with
(a) An observer at rest in some inertial frame receives a photon that was emitted in direction n by an atom moving with ordinary velocity v (Fig. 2.6). The photon frequency and energy as measured by
Derive Eq. (2.47b) by the same method as was used to derive Eq. (2.47a). Then show, by a geometric, frame-independent calculation, that Eq. (2.47b) implies Eq. (2.47a). E = Faß wB, B² = ¹
Consider the 4-dimensional parallelepiped V whose legs arewhere (t , x, y, z) = (x0, x1, x2, x3) are the coordinates of some inertial frame. The boundary ∂V of this V has eight 3-dimensional
In Minkowski spacetime, the set of all events separated from the origin by a time like interval a2 is a 3-surface, the hyperboloid t2 − x2 − y2 − z2 = a2, where {t , x, y, z} are Lorentz
Show, using spacetime diagrams and also using frame-independent calculations, that the law of conservation of 4-momentum forbids a photon to be absorbed by an electron, e + γ → e, and also forbids
In a long-ago era when an airline named Trans World Airlines (TWA) flew around the world, Josef Hafele and Richard Keating (1972a) carried out a real live twins paradox experiment: They synchronized
Consider the global law of charge conservation ∫∂V Jαd∑α = 0 for a special choice of the closed 3-surface ∂V: The bottom of ∂V is the ball {t = 0, x2 + y2 + z2 ≤ a2}, where {t , x, y,
In Newtonian theory, the gravitational potential Φ exerts a force F = dp/dt = −m∇Φ on a particle with mass m and momentum p. Before Einstein formulated general relativity, some physicists
Show that Eq. (2.19) can be true for all time like, unit-length vectors u(vector) if and only if F is antisymmetric. F(u, ū) = 0. (2.19)
Do Ex. 1.15 in Chap. 1.Data from Exercises 1.15 of Chapter 1Convert the following equations from the geometrized units in which they are written to SI units.(a) The “Planck time” tP expressed in
Complete the derivation of the invariance of the interval given in Box 2.4, using the Principle of Relativity in the form that the laws of physics must be the same in theprimed and unprimed frames.
Derive the relativistic component manipulation rules (2.23e)–(2.23g). [Contravariant components of T(____)S_₁_)] = Tage (2.23e)
Without introducing any coordinates or basis vectors, show that when a particle with charge q interacts with electric and magnetic fields, its kinetic energy changes at a rate dE/dt =qv. E. (1.8)
Consider a particle moving in a circle with uniform speed v = |v| and uniform magnitude a = |a| of acceleration. Without introducing any coordinates or basis vectors, do the following.(a) At any

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