Questions and Answers of Modern Classical Physics

A good illustration of the importance of the Pv term in the energy flux is provided by the Joule-Kelvin method commonly used to cool gases (Fig. 13.8). Gas is driven from a high-pressure chamber 1
There’s a hole in my bucket. How long will it take to empty? (Try an experiment, and if the time does not agree with the estimate, explain why not.)
When dealing with differential equations describing a physical system, it is often helpful to convert to dimensionless variables. Polytropes (nonrotating, spherical fluid bodies with the polytropic
As mountaineers know, it gets cooler as you climb. However, the rate at which the temperature falls with altitude depends on the thermal properties of air. Consider two limiting cases.(a) In the
Fill in the details of the analysis of time-averaged seeing in Box 9.2.More specifically, do the following. If you have difficulty, Roddier (1981) may be helpful.(a) Give an order-of-magnitude
X-rays with wavelength 8.33˚A (0.833 nm) coming from a point source can be reflected at shallow angles of incidence from a plane mirror. The direct ray from a point source to a detector 3m away
Derive Eq. (7.94). M = (0²/+ 1/B²) (0²/+8²)¹/²B (7.94)
Consider hydrogen gas in statistical equilibrium at a temperature T ec2/kB ≈ 6 × 109 K. Electrons at the high-energy end of the Boltzmann energy distribution can produce electron-positron pairs by
(a) Consider H2O in contact with a heat and volume bath with temperature T and pressure P. For certain values of T and P the H2O will be liquid water; for others, ice; for others, water vapor—and
For a nonrelativistic, classical, ideal gas (no interactions between particles), evaluate the statistical sum (5.50) to obtain G(P , T , N), and from it deduce the standard formula for the ideal-gas
In Sec. 5.4.1, we explained the experimental meaning of the free energy F for a system in contact with a heat bath so its temperature is held constant, and in Ex. 5.5h we did the same for contact
In Part V, when studying fluid dynamics, we shall encounter an adiabatic index[Eq. (13.2)] that describes how the pressure P of a fluid changes when it is compressed adiabatically (i.e., compressed
Prove the properties of entropy enumerated in Sec. 4.11.4.Data from Sec. 4.11.4Because of the similarity of the general formulas for information and entropy (both proportional to ∑n −pn ln pn),
By using more accurate approximations to Eq. (4.48a), explore the onset of the condensation near T = Tc0 . More specifically, do the following.(a) Approximate the numerator in Eq. (4.48a) by q2 + 3q,
(a) When the early universe was ∼200 s old, its principal constituents were photons, protons, neutrons, electrons, positrons, and (thermodynamically isolated) neutrinos and gravitons. The photon
We will study classical sound waves propagating through an isotropic, elastic solid. As we shall see, there are two types of sound waves: longitudinal with frequency-independent speed CL, and
We have described distribution functions for particles and photons and the forms that they have in thermodynamic equilibrium. An extension of these principles can be used to constrain the manner in
The GPS satellites are in circular orbits at a height of 20,200 km above Earth’s surface, where their orbital period is 12 sidereal hours. If the ticking rates of the clocks on the satellites were
Consider a weak, planar gravitational wave propagating in the z direction, written in a general Lorenz gauge [Eqs. (27.19)]. Show that by appropriate choices of new gauge-change generators that have
Gravitational waves from a distant source travel through the Sun with impunity (negligible absorption and scattering), and their rays are gravitationally deflected. The Sun is quite centrally
Show that conditions (i), (ii), and (iii) preceding Eq. (27.48) guarantee that the multipolar expansion of the gravitational-wave fields will have the form (27.48).ConditionsThe wave fields h+ and
Consider a mass m attached to a spring, so it oscillates along the z-axis of a Cartesian coordinate system, moving along the world line z = a cos Ωt, y = x = 0. Use the quadrupole-moment formalism
Extrapolating Eqs. (27.71)–(27.73) into the strong-gravity regime, estimate the maximum gravitational- wave amplitude and emitted power for a nonspinning binary black hole with equal masses and
Many precision tests of general relativity are associated with binary pulsars in elliptical orbits.(a) Verify that the radius of the relative orbit of the pulsars can be written as r = p/(1 + e cos
We have hitherto focused on the statistical properties of the cosmological perturbations as probed by a variety of observations. However, we on Earth occupy a unique location in a specific
Derive the dispersion relation ω2(k) for axisymmetric perturbations of the Θ-pinch configuration when the magnetic field is confined to the cylinder’s exterior, and conclude from it that the
Consider a particle that is at rest in the TT coordinate system of the gravitational-wave metric (27.80) before the gravitational wave arrives. In the text it is shown that the particle’s
Wave your arms rapidly, and thereby try to generate gravitational waves.(a) Using classical general relativity, compute in order of magnitude the wavelength of the waves you generate and their
(a) Derive the behavior [Eq. (27.31)] of h+ and h× under rotations in the transverse plane.(b) Show that, with the orientations of spatial basis vectors described after Eq. (27.31), h+ and h× are
(a) One possible choice of slices of simultaneity for Schwarzschild spacetime is the set of 3-surfaces {t = const}, where t is the Schwarzschild time coordinate. Show that the unique family of
(a) Show that, as the surface of an imploding star approaches R = 0, its world line in Schwarzschild coordinates asymptotes to the curve {(t , θ , ∅) = const, r variable}.(b) Show that this curve
Not surprisingly, there are several other approaches to deriving the possible forms of ∑(χ). Another derivation exploits the symmetries of the Riemann tensor.(a) The 3-dimensional Riemann
Consider the triangle formed by the three geodesics in Fig. 28.3. In a flat space, the exterior angle ζ must equal θ + ψ. However, if the space is homogeneous and positively curved, then the angle
Suppose that the universe contained a significant component in the form of isotropic but noninteracting particles with momentum p and rest mass m. Suppose that they were created with a distribution
Astronomers find it convenient to use the redshift z = 1/a − 1 to measure the size of the universe when the light they observed was emitted.(a) Perform a Taylor expansion in z to show that the
Assume that a fraction ∼0.2 of the baryons in the universe is associated with galaxies, split roughly equally between stars and gas. Also assume that a fraction ∼10−3 of the baryons in each
Make a simple (numerical)model of a spherical galaxy in which the dark matter particles moving in the (Newtonian) gravitational field they create behave like collisionless plasma particles moving in
Calculate three contributions to the pressure of the contemporary universe.(a) Baryons. Assume that most of the baryons in the universe outside of stars make up a uniform, hot intergalactic medium
(a) Estimate the minimum fraction of the rest mass energy of the hydrogen that must have undergone nuclear reactions inside stars to have ionized the remaining gas when a ∼ 0.1.(b) Suppose that
Explore nonlinear effects in the growth of perturbations in the gravitational age— when radiation and the cosmological constant can be ignored—by considering the evolution of a sphere in which
Neutrinos have mass, which becomes measurable at late times through its influence on the growth of structure.(a) Explain how the expansion of the universe is changed if there is a single dominant
Assume that the universe will continue to expand according to Eq. (28.43).(a) Calculate the behavior of the angular diameter distance and the associated volume as a function of the scale factor for
Rather surprisingly, it turns out that a certain type of supernova explosion (called “Type 1a” and associated with detonating white-dwarf stars) has a peak luminosity L that can be determined by
There are many ways to represent the polarization of electromagnetic radiation. A convenient one that is used in the description of CMB fluctuations was introduced by Stokes.(a) Consider a
The precision with which the low-l spherical harmonic power spectrum can be determined observationally is limited because of the low number of independent measurements that can be averaged over. Give
We have explained how the peaks in the CMB temperature fluctuation spectrum arise because the sound waves all began at the same time and are all effectively observed at the same time, while they
Consider an extended congruence propagating through an otherwise homogeneous universe, from which all matter has been removed. Show that the affine distance functions as an effective angular diameter
Consider a single light ray propagating across the universe from a source at log a = −0.5 to us. The cumulative effect of all the deflections caused by large-scale inhomogeneities makes the
A blind (but hearing) cosmologist observed the radiation-dominated universe. He detected faint tones and noted that their frequencies declined as t−1/2 and believed (correctly) that the sound speed
We have made many simplifying assumptions in this chapter to demonstrate the strong connection to the principles and techniques developed in the preceding 27 chapters. It is possible to improve on
There are many elaborations of standard cosmology either involving new features following from known physics or involving new physics. While no convincing evidence exists for any of the mas of this
We explore the structure of the wake behind the cylinder when the Reynolds number is high enough that the flow is turbulent. For comparison, here we compute the wake’s structure at lower Reynolds
One often hears the claim that water in a bathtub or basin swirls down a drain clockwise in the northern hemisphere and counterclockwise in the southern hemisphere. In fact, on YouTube you are likely
Derive results (i), (ii), and (iii) in the last paragraph of Box 14.4.Box 14.4. BOX 14.4. STREAM FUNCTION FOR A GENERAL, TWO-DIMENSIONAL, INCOMPRESSIBLE FLOW ™Z Consider any orthogonal coordinate
Place tea leaves and water in a tea cup, glass, or other larger container. Stir the water until it is rotating uniformly, and then stand back and watch the motion of the water and leaves. Notice that
Verify that for the constant-angular-momentum flow of Fig. 14.1b, with v = j × x/ω̅2, two neighboring fluid elements move around each other with angular velocity +j/ω̅2 when separated
How much more would you weigh in a vacuum?
Use Archimedes’ law to explain qualitatively the conditions under which a boat floating in still water will be stable to small rolling motions from side to side. You might want to define and
Consider a stationary, axisymmetric planet, star, or disk differentially rotating under the action of a gravitational field. In other words, the motion is purely in the azimuthal direction.(a)
For the microcanonical ensemble considered in this section, derive Eq. (5.5) for the pressure using a thought experiment involving a pressure-measuring device. p = - ᎧᏋ ( 5 ) . ᎯᏙ S, N (5.5)
For the fold caustic discussed in the text, assume that the phase change introduced by the imperfect lens is nondispersive, so that the φ(ã, x̃) in Eq. (8.45) satisfies φ ∝ λ−1. Show that
Consider a plane, monochromatic electromagnetic wave with angular frequency ω, whose electric field is expressed in terms of its complex amplitude X1 + iX by Eq. (10.58). Because the field
(a) Derive Eq. (10.68) for the polarization induced in an isotropic medium by a linearly polarized electromagnetic wave.(b) Fill in the remaining details of the derivation of Eq. (10.69) for the
In some inertial reference frame, the vector A(vector) and second-rank tensor T have as their only nonzero components A0 = 1, A1 = 2; T00 = 3, T01 = T10 = 2, T11 = −1. Evaluate T (A(vector),
At low temperatures certain fluids undergo a phase transition to a superfluid state. A good example is 4He, for which the transition temperature is 2.2 K. As a superfluid has no viscosity, it cannot
(a) Show that the spatially variable part of the gravitational potential for a uniform density, nonrotating planet can be written as Φ = 2πGρr2/3, where ρ is the density.(b) Hence argue that the
Suppose that a spherical bubble has just been created in the water above the hydrofoil in the previous exercise. Here we analyze its collapse—the decrease of the bubble’s radius R(t) from its
Consider the pulsatile flow of blood through one of the body’s larger arteries. The pressure gradient dP/dz = P'(t) consists of a steady term plus a term that is periodic, with the period of the
Integrate the energy density U of Eq. (5) of Box 13.4 over the interior and surroundings of an isolated gravitating system to obtain the system’s total energy. Show that the gravitational
Consider two stars with the same mass M orbiting each other in a circular orbit with diameter (separation between the stars’ centers) a. Kepler’s laws tell us that the stars’ orbital angular
(a) Consider steady flow of an ideal fluid. The Bernoulli function (13.51) is conserved along streamlines. Show that the variation of B across streamlines is given by Crocco’s theorem:(b) As an
(a) Derive the Lagrangian equation (13.75) for the rate of increase of entropy in a dissipative fluid by carrying out the steps in the sentence preceding that equation.(b) From the Lagrangian
A hydrofoil moves with speed V at a depth D = 3m below the surface of a lake; see Fig. 13.7. Estimate how fast V must be to make the water next to the hydrofoil boil. This boiling, which is called
(a) Show that in the nonrelativistic limit, the components of the perfect-fluid stress energy tensor (13.85) take on the forms (13.91), and verify that these agree with the densities and fluxes of
A viscous fluid flows steadily (no time dependence) in the z direction, with the flow confined between two plates that are parallel to the x-z plane and are separated by a distance 2a. Show that the
Estimate the collision mean free path of the air molecules around you. Hence verify the estimate for the kinematic viscosity of air given in Table 13.2. TABLE 13.2: Approximate kinematic viscosity
By manipulating the differential forms of the law of rest-mass conservation and the law of energy conservation, derive the constancy of B = (ρ + P)γ/ρo along steady flow lines, Eq. (13.88).
By taking the curl of the Euler equation (13.44), derive the vorticity evolution equation (14.9) for a compressible, barotropic, inviscid flow. dv dt = Əv at + (v. V)v: VP P +g for an ideal
(a) Figure 14.5 shows photographs of two particularly destructive tornados and one waterspout (a tornado sucking water from the ocean). For the tornados the wind speeds near the ground are
Consider a velocity field with non vanishing curl. Define a locally orthonormal basis at a point in the velocity field, so that one basis vector, ex, is parallel to the vorticity. Now imagine the
Explain why the pressure and temperature of the core of a wingtip vortex are significantly lower than the pressure and temperature of the ambient air. Under what circumstances will this lead to
At time t = 0, a 2-dimensional barotropic flowhas a velocity field, in circular polar coordinates, v = (j/ω̅)e∅ (Fig. 14.1b); correspondingly, its vorticity is ω = 2πjδ(x)δ(y)ez: it is a
Give an expression for the change in the thrust—the momentum crossing a surface perpendicular to the tube per unit time—along a slender stream tube when the discharge and power are conserved.
Smoke rings (ring-shaped vortices) blown by a person (Fig. 14.8a) propagate away from him. Similarly, a hovering hummingbird produces ring-shaped vortices that propagate downward (Fig. 14.8b). Sketch
Rooms are sometimes heated by radiators (hot surfaces) that have no associated blowers or fans. Suppose that, in a room whose air is perfectly still, a radiator is turned on to high temperature. The
Sketch the streamlines for the following stationary 2-dimensional flows, determine whether the flow is compressible, and evaluate its vorticity. The coordinates are Cartesian in parts (a) and (b),
The north Atlantic Ocean exhibits the pattern of winds and ocean currents shown in Fig. 14.18. Westerly winds blow from west to east at 40° latitude. Trade winds blow from east to west at 20°
Insert gravity into the analysis of the Kelvin-Helmholtz instability (with the uniform gravitational acceleration g pointing perpendicularly to the fluid interface, from the upper “+” fluid to
Consider stationary incompressible flow around a cylinder of radius a with sufficiently large Reynolds number that viscosity may be ignored except in a thin boundary layer, which is assumed to extend
Fill a bathtub with water and sprinkle baby powder liberally over the water’s surface to aid in viewing the motion of the surface water. Then take a spatula, insert it gently into the water, move
(a) Fill in the details of the dimensional-analysis derivation of the Kolmogorov spectrum (15.27) for a quantity such as n that is transported by the fluid and thus satisfies dn/dt = 0. In
Consider low-Reynolds-number flow past an infinite cylinder whose axis coincides with the z-axis. Try to repeat the analysis we used for a sphere to obtain an order-of magnitude estimate for the drag
Episodic glaciation subjects Earth’s crust to loading and unloading by ice. The last major ice age was 10,000 years ago, and the subsequent unloading produces a nontidal contribution to the
Estimate the Reynolds numbers for the following flows. Make sketches of the flow fields, pointing out any salient features.(a) A hang glider in flight.(b) Plankton in the ocean.(c) A physicist waving
An auto manufacturer wishes to reduce the drag force on a new model by changing its design. She does this by building a one-sixth scale model and putting it into a wind tunnel. How fast must the air
Consider an inviscid (ν = 0), incompressible flow near a plane wall where a laminar boundary layer is established. Introduce coordinates x parallel to the wall and y perpendicular to it. Let the
A well-hit golf ball travels about 300 yards. A fast bowler or fastball pitcher throws a cricket ball or baseball at more than 90mph (miles per hour). A table-tennis player can hit a forehand return
Compute the width w(x) and velocity deficit uo(x) for the 3-dimensional turbulent wake behind a sphere.
Consider the logistic equation (15.35) for the special case a = 1, which is large enough to ensure that chaos has set in.(a) Make the substitution xn = sin2πθn, and show that the logistic equation
Use a computer to calculate the first five critical parameters aj for the sequence of numbers generated by the logistic equation (15.35). Hence verify that the ratio of successive differences tends

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