Mechanics

Problem 1.1. A Wham-O Super-Ball is a hard spherical ball of radius a. The bounces of a Super-Ball on a surface with friction are essentially elastic and non-slip at the point of contact. How should you throw a Super-Ball if you want it to bounce back and forth as shown in Figure 1.1? (Super-Ball is a registered trademark of Wham-O Corporation, San Gabriel, California.)

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Problem 1.2. Suppose a spacecraft of mass mo and cross-sectional area A is coasting with velocity vo when it encounters a stationary dust cloud of density ρ. Solve for the subsequent motion of the spacecraft assuming that the dust sticks to its surface and that A is constant over time.

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Problem 1.3. The science fiction writer R. A. Heinlein describes a "skyhook" satellite that consists of a long rope placed in orbit at the equator, aligned along a radius from the center of the earth, and moving so that the rope appears suspended in space above a fixed point on the equator (Figure 1.2). The bottom of the rope hangs free just above the surface of the earth (radius R). Assuming that the rope has uniform mass per unit length (and that the rope is strong enough to resist breaking!), find the length of the rope.

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Problem 1.4. Three identical objects of mass m are connected by springs of spring constant k, as shown in Figure 1.3. The motion is confined to one dimension. At t = 0, the masses are at rest at their equilibrium positions. Mass A is then subjected to an external driving force,

F(t) = f cos ωt, for t > 0. (1.1)

Calculate the motion of mass C.

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Problem 1.5. A uniform density ball rolls without slipping and without rolling friction on a turntable rotating in the horizontal plane with angular velocity Ω (Figure 1.4). The ball moves in a circle of radius r centered on the pivot of the turntable. Find the angular velocity a; of motion of the ball around the pivot.

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Problem 1.6. A blob of putty of mass m falls from height h onto a massless platform which is supported by a spring of constant k. A dashpot provides damping. The relaxation time of the putty is short compared to that of putty-plus-platform: the putty instantaneously hits and sticks.

a) Sketch the displacement of the platform as a function of time, under the given initial conditions, when the platform with putty attached is critically damped.

b) Determine the amount of damping such that, under the given initial conditions, the platform settles to its final position the most rapidly without overshoot.

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Problem 1.7. A mass m slides on a horizontal frictionless track. It is connected to a spring fastened to a wall. Initially, the amplitude of the oscillations is A1 and the spring constant is k1. The spring constant then decreases adiabatically at a constant rate until the value k2 is reached. (For example, suppose the spring is being dissolved by nitric acid.) What is the new amplitude?

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Problem 1.8. A soap film is stretched between two coaxial circular rings of equal radius R. The distance between the rings is d. You may ignore gravity. Find the shape of the soap film.

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Problem 1.9. A bead of mass m slides without friction on a circular loop of radius a. The loop lies in a vertical plane and rotates about a vertical diameter with constant angular velocity ω (Figure 1.5).

a) For angular velocity ω greater than some critical angular velocity ωc, the bead can undergo small oscillations about some stable equilibrium point θo. Find ωc and θo(ω).

b) Obtain the equations of motion for the small oscillations about θo as a function of ω and find the period of the oscillations.

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Problem 1.10. If the solar system were immersed in a uniformly dense spherical cloud of weakly-interacting massive particles (WIMPs), then objects in the solar system would experience gravitational forces from both the sun and the cloud of WIMPs such that

Fr = - k/r2 - br. (1.2)

Assume that the extra force due to the WIMPs is very small (i.e., b<< k/r3).

a) Find the frequency of radial oscillations for a nearly circular orbit and the rate of precession of the perihelion of this orbit.

b) Describe the shapes of the orbits when r is large enough so that Fr ≈ -br.

Electricity & Magnetism

Problem 2.1. A conductor at potential V = 0 has the shape of an infinite plane except for a hemispherical bulge of radius a (Figure 2.1). A charge q is placed above the center of the bulge, a distance p from the plane (or p - a from the top of the bulge). What is the force on the charge?

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Problem 2.2. A "tenuous plasma" consists of free electric charges of mass m and charge e. There are n charges per unit volume. Assume that the density is uniform and that interactions between the charges may be neglected. Electromagnetic plane waves (frequency ω, wave number k) are incident on the plasma.

a) Find the conductivity σ as a function of ω.

b) Find the dispersion relation — i.e., find the relation between k and ω.

c) Find the index of refraction as a function of ω. The plasma frequency is defined by ω2p [equivalent to] 4πne2/m, if e is expressed in esu. What happens if ω < ωp?

d) Now suppose there is an external magnetic field B0. Consider plane waves traveling parallel to B0. Show that the index of refraction is different for right- and left-circularly polarized waves. (Assume that the magnetic field of the traveling wave is negligible compared to B0.)

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Problem 2.3. A cylindrical resistor (Figure 2.2) has radius b, length L, and conductivity σ1. At the center of the resistor is a defect consisting of a small sphere of radius a inside which the conductivity is σ2. The input and output currents are distributed uniformly across the flat ends of the resistor.

a) What is the resistance of the resistor if σ1 = σ2?

b) Estimate the relative change in the resistance to first order in σ1 - σ2 if σ1 ≠ σ2. (Make any assumptions needed to simplify your method of estimation.)

c) Suppose L -> ∞ and b -> ∞, but a uniform current density j0 continues to flow across the ends of the resistor. Calculate the current density inside the spherical defect.

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Problem 2.4. A thin, straight, conducting wire is centered on the origin, oriented along the z-axis, and carries a current I = zI0 cos ω0t everywhere along its length l. Define λ0 [equivalent] 2πcc/ω0.

a) What is the electric dipole moment of the wire?

b) What are the scalar and vector potentials everywhere outside the source region (at a distance r >> l)? State your choice of gauge and make no assumption about the size of λ0.

c) Consider the potentials in the regime r >> l >> λ0. Qualitatively describe the radiation pattern and compare it to the standard dipole case, where r >> λ0 >> l.

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Problem 2.5. As shown in Figure 2.3, a wheel consisting of a large number of thin conducting spokes is free to rotate about an axle. A brush always makes electrical contact with one spoke at a time at the bottom of the wheel. A battery with voltage V feeds current through an inductor, into the axle, through a spoke, to the brush. A permanent magnet provides a uniform magnetic field B into the plane of the paper. At time t = 0 the switch is closed, allowing current to flow. The radius and moment of inertia of the wheel are R and J repectively. The total inductance of the current path is L, and the wheel is initially at rest. Neglecting friction and resistivity, calculate the battery current and the angular velocity of the wheel as functions of time.

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Problem 2.6. A right-circular cylinder of radius R, length L, and uniform mass density ρ has a uniform magnetization M parallel to its axis. If it is placed below an infinitely-permeable flat surface, it is found to stick for some lengths L >> R. What is the maximum length L such that the magnetic force prevents the cylinder from falling due to gravity?

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Problem 2.7. Consider an infinitely long transmission line which consists of lumped circuit elements as shown in Figure 2.4. Find the dispersion relation (ω versus λ) for periodic waves traveling down this line. What is the cut-off frequency?

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Problem 2.8. In 1890, 0. Wiener carried out an experiment which may be said to have photographed electromagnetic waves (Figure 2.5).

a) A plane wave is normally incident on a perfectly reflecting mirror. A glass photographic plate is placed on the mirror so that it makes a small angle α to the mirror. The photographic emulsion is almost transparent. But when it is later developed, a striped pattern is found due to the action of the wave. Predict the position and spacing of the black stripes which appear on the developed "negative" plate. Ignore any attenuation or reflection due to the glass photographic plate itself.

b) The experiment is repeated for incident waves making angle 45° to the normal to the mirror. Now what is the pattern of blackening on the negative? Distinguish the cases of light polarized with E parallel and perpendicular to the plane of incidence (i.e., the scattering plane).

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Problem 2.9. An infinitely long, thin-walled circular cylinder of radius b is split into two half cylinders. The upper half is fixed at potential V = + V0 and the lower half at V = -V0.

a) Find the potential inside and outside the cylinder.

b) Calculate the charge density as a function of φ. (If your answer is in the form of an infinite sum, calculate this sum.)

c) Find the capacitance per unit length of the device when the two half cylinders are a distance ε apart at the rims ([member of] << b),

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Problem 2.10. An electron is released from rest at a large distance r0 from a nucleus of charge Ze and then "falls" toward the nucleus. For what follows, assume that the electron's velocity is such that v<< c and that the radiative reaction force on the electron is negligible.

a) What is the angular distribution of the emitted radiation?

b) How is the emitted radiation polarized?

c) What is the radiated power as a function of the separation between the electron and the nucleus?

d) What is the total energy radiated by the time the electron is a distance r< r0 from the nucleus?

Quantum Mechanics

Problem 3.1. A particle of mass m interacts in three dimensions with a spherically symmetric potential of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.1)

In other words, the potential is a delta function that vanishes unless the particle is precisely a distance a from the center of the potential. Here c is a positive constant.

a) Find the minimum value of c for which there is a bound state.

b) Consider a scattering experiment in which the particle is incident on the potential with a low velocity. In the limit of small incident velocity, what is the scattering cross-section? What is the angular distribution?

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Problem 3.2. A particle of mass M bounces elastically between two infinite plane walls separated by a distance D. The particle is in its lowest possible energy state.

a) What is the energy of this state?

b) The separation between the walls is slowly (i.e., adiabatically) increased to 2D.

i) How does the expectation value of the energy change?

ii) Compare this energy change with the result obtained classically from the mean force exerted on a wall by the bouncing ball.

c) Now assume that the separation between the walls is increased rapidly, with one wall moving at a speed >> [square root of E/M]. Classically there is no change in the particle's energy since the wall is moving faster than the particle and cannot be struck by the particle while the wall is moving.

i) What happens to the expectation value of the energy quantum-mechanically?

ii) Compute the probability that the particle is left in its lowest possible energy state.

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Problem 3.3. Consider a particle of charge e and mass m in constant, crossed E and B fields:

E = (0,0, E), B = (0, B, 0), r = (x, y, z). (3.2)

a) Write the Schrödinger equation (in a convenient gauge).

b) Separate variables and reduce it to a one-dimensional problem.

c) Calculate the expectation value of the velocity in the x-direction in any energy eigenstate (sometimes called the drift velocity).

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Problem 3.4. A particle of mass m and charge q sits in a harmonic oscillator potential V = k (x2 + y2 + z2)/2. At time t = -∞ the oscillator is in its ground state. It is then perturbed by a spatially uniform time-dependent electric field

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.3)

where A and τ are constants. Calculate in lowest-order perturbation theory the probability that the oscillator is in an excited state at t = ∞.

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Problem 3.5. Consider an elastic scattering experiment a + X ->a + X with a and X having zero spin, and X much heavier than a. The total cross-section σtot as a function of momentum hk behaves as shown in Figure 3.1. A contribution to σtot from a resonance is observable at all angles except 90°, where the contribution vanishes. Far off resonance σtot is isotropic.

a) What is the angular momentum J of the resonance?

b) Calculate the approximate value of the differential cross-section at resonance at a scattering angle of 180°.

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Problem 3.6. a) A spin 1/2 electron is in a uniform magnetic field B0 = Boz. At time t = 0 the spin is pointing in the x-direction, i.e., <Sx (t = 0)) = h/2. Calculate the expectation value t)> at time t.

b) An additional magnetic field B1 = 1/2 B1[cos(ωt)x + sin(ωt)y] is now applied. If an electron in the combined field B0 + B1 has spin pointing along +z at time t = 0, what is the probability that it will have flipped to -z at time t?

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Problem 3.7. Pion-nucleon scattering at low energies can be qualitatively described by an effective interaction potential of the form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.4)

Here a and μ, are constants, r is the relative pion-nucleon coordinate, and I([??]) and I(N) are the pion and nucleon isospin operators.

a) Calculate the ratio of the scattering cross-sections with total isospin I = 3/2 and I = 1/2.

b) Calculate in the Born approximation the low-energy total crosssections for the reactions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

NB: If you are not familiar with isospin, you may consider the two particles to have (ordinary) spin 1 and spin 1/2 with a spin-spin interaction and initial and final states which are eigenstates of Sz for each particle. The corresponding Sz values are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.5)