Download Advanced Physics Problems: Mechanics, Thermodynamics, Statistical Mechanics, and Relativit and more Summaries Classical Physics in PDF only on Docsity!
August 2019
CLASSICLAL TEST BANK
- Carnot Cycle Efficiency [500 level] Consider a Carnot cycle using an ideal gas with constant specific heat CV as the working substance. (a) Specify all segments of the path in P - V space that make up the cycle. (b) Find the work done on the adiabatic legs. (c) Find the work done and heat transfered on the isothermal legs. (d) Determine the efficiency.
- Lagrangian Mechanics [700 level] A small object of mass m slides without friction on the inside surface of a cone. The axis of the cone is vertical, with the vertex at the bottom and cone angle θ. (a) Find the Lagrangian. (b) Find the equations of motion. (c) Identify all conserved quantities.
- Small Oscillations [500 level] A mass m slides frictionlessly on the upper surface of a wedge of mass M and angle θ, which in turn slides frictionlessly on a horizontal surface. The mass m is connected to the top of the wedge by a spring, with spring constant k.
m
k
M
(a) Find the Lagrangian for this system. (b) Calculate the frequency of small oscillations about the equilibrium point of the system.
- Newtonian Mechanics [500 level] An object of mass 1 kg moving vertically downward in a uniform gravitational field is subject to a damping force proportional to its velocity (damping force constant 0.5 N/m/s). (a) If its initial velocity is forty times its terminal velocity, what is the time
required to reach 1.1 times terminal velocity? (b) What is the energy dissipated during the slowdown to 1.1 times terminal velocity?
- Binding Energy [500 level] (a) How much energy would you have to supply to break up the Earth com- pletely, i.e. convert it into an infinitely dispersed dust? You can ignore: (i) any and all technical hurdles involved in this process (that’s for the engineers to worry about); (ii) chemistry and radiation losses; (iii) any nonuniformity in the Earth’s composition. (b) If you reassembled it rapidly (so that no energy radiated, and ignoring chemistry, etc) would it melt? Some numbers: MC “ 6 ˆ 1024 kg TCmelt “ 1 ˆ 104 K RC “ 6 ˆ 106 m G “ 6 ˆ 10 ´^11 N m^2 {kg^2 CC “ 1 ˆ 103 J{kg K
- Small Oscillations [500 level] A pipe of radius r rolls without slipping inside another of radius R. The outer pipe is fixed in position. (a)Find the frequency of small oscillations in which the axes remain parallel. (b) Discuss the limits as (i) r Ñ 0 and (ii) r Ñ R.
- Newtonian Mechanics [500 level] A satellite attached to the surface of the Earth by a long flexible cable with linear mass density μ, revolves with constant angular velocity ω, equal to the rotation frequency of the Earth. (a) Show that the cable must be straight and radial. (b) Derive and solve the differential equation for the tension in the cable as a function of r, the satellite’s distance from the center of the Earth. Assume the cable is long compared to the Earth’s radius. (c) Demonstrate that the tension is maximum at
r “
gR^2 C ω^2
where g is the acceleration due to gravity at the Earth’s surface.
- Newtonian Mechanics [500 level] (a) If all linear dimensions of the solar system were doubled, and if the masses of everything in the system were also doubled at the same time, how would the orbital period of the Moon compare to its current value? (b) By what factor would the total energy change?
- Lagrangian Dynamics [700 level]
m
L
m^ L
(a) For the double pendulum shown, find the kinetic energy in terms of the angular variables and their derivatives. (b) Deduce the potential energy. (c) What is the Lagrangian? Find the equations of motion in the angular variables. (d) Given small oscillations, equal pendula arm lengths, and equal masses, solve the equations of motion.
- Newtonian Mechanics [500 level] A rocket initially at rest is fired vertically upward in a uniform gravitational field. The rocket ejects mass, from an initial propellant mass Mp at a muzzle velocity vp at a rate of α. (a) Calculate the minimum α required for the rocket to lift off the ground. (b) For this α, calculate the maximum speed of the rocket.
- Thermodynamics [500 level] A house is to be maintained at a temperature lower than the external tem- perature Tout. For convenience, consider a cubical house of side length L. It is cooled by a continuously running air conditioning unit, which is a perfect Carnot engine with constant input power P. The wall thickness is W and their thermal conductivity is k. (a) Calculate the inside temperature. (b) What happens when the insulation is poor or the power is low?
- Action [700 level] (a) Write down a formula for the period T of a particle executing one-dimensional periodic motion in a potential.
(b) Show that this period T is the derivative of the action
I “
dx ppxq
with respect to the energy E. (c) Calculate the period of a particle of mass m and energy E in a potential
U “ α |x|.
- Ideal Gas [500 level] A spherical soap bubble with surface tension (energy per unit surface area) Σ is filled with N molecules of an ideal gas. Outside the bubble is vacuum, and you may neglect radiative heat loss. (a) What is the equilibrium radius of the bubble as a function of temperature? (b) If the bubble has an initial small radius, R « 0, how much energy must be supplied in the form of heat to double its radius?
- Small Oscillations [500 level] A bead of mass m slides without friction on a circular loop of radius R. The loop lies in a vertical plane and rotates about its vertical diameter with con- stant angular speed ω. Let θ be the angular position of the bead along the loop.
m
R
(a) Show that there is a stable equilibrium at θ “ 0 if ω ă ωc for some critical
- Newtonian Mechanics [500 level] Two particles, m 1 and m 2 , attract each other with a central force Fprq “ ´kr, where r is the separation between the particles. Let V pr “ 0 q “ 0, with E being the internal energy and L the relative angular momentum. (a) Given L, what is the allowed range for E? (b) Given E and L, what are the minimal and maximal distances the particles may be from one-another?
- Hamiltonian Mechanics [700 level] Consider a system described by the general coordinates qi and pi. If for some conjugate pair qr and pr, both qr and pr are confined to finite ranges, then the virial theorem holds (^) B BH Bpr
pr
F
B
qr
BH
Bqr
F
You will prove that this is so. (a) Establish a simple relation between (^) dtd pprqrq and (^) BBHpr pr ´ qr B BHqr. (b) Consider the average of the relation from (a) over long times, and thereby prove the virial theorem. Recall that
xf y “ lim T Ñ
T
ż (^) T
0
dt f ptq.
(c) Use the virial theorem to show that for a three-dimensional harmonic oscillator, xT y “ xV y, where T and V are the kinetic and potential energies, respectively.
- Newtonian Mechanics [500 level] A homogeneous cube of length L is balanced on one edge on a horizontal plane. If the cube is given a slight nudge so that it falls, show that the angular speed of the plane when the face strikes the plane is
ω^2 “ α
g L
p
2 ´ 1 q,
and find α for the cases of: (a) no slipping (b) slipping without friction.
- Normal Modes [500 level] Consider the system of masses and springs shown in the figure, with Hookian springs, one-dimensional motion, and no friction.
m 2m
k k k
(a) Find the equations of motion in terms of x 1 and x 2 (the positions of the masses, relative to their equilibrium positions) and find the normal mode fre- quencies. (b) Without further calculation: (i) Give an argument showing that the general motion of the system is a linear combination of the normal modes. (ii) Describe qualitatively the motions of the two masses in each of the normal modes. (iii) Describe initial conditions that result in only one normal mode being excited, so that the motion can be described by a single frequency.
- Newtonian Mechanics [500 level] A uniform, thin square plate ABCD on a frictionless horizontal surface is rotating around its vertical diagonal AC with angular speed ω, as shown in the figure.
A
B
C
D
(a) If the vertex B is suddenly stopped, in which direction and with what angular speed does the square rotate around B? (b) Discuss the conservation (or non-conservation) of mechanical energy in the process given in (a).
- Hamiltonian Mechanics [500 level]
- Orbital Mechanics [700 level] (a) Consider an attractive central force Fprq “ ´pk{rnqˆr. Given k and n, find the condition that guarantees the existence of a stable circular orbit. (b) Find a similar condition for a general attractive central force Fprq. (c) Investigate the stability of circular orbits in a force field described by the screened Coulomb potential
V prq “ ´
k r
e´r{a,
with positive constants k and a.
- Hamiltonian Mechanics [700 level] (a) For a one-dimensional harmonic oscillator, write down Hamilton’s equa- tions of motion and show that they agree with Newton’s equations of motion. (b) Explain the behavior of their solutions in phase space.
- Statistical Mechanics [700 level] Consider an ideal gas with N molecules, molecular mass m, volume V , and temperature T. (a) What are the partition function, the Helmholtz free energy, and the chem- ical potential for this system? (b) Let the gas be in contact with a large absorption surface with NA absorp- tion sites, each of which can absorb at most one molecule. The absorption energy per molecule is EA. Assuming N " NA, what is the grand partition function for the absorption sites? (c) What fraction of the sites are occupied?
- Statistical Mechanics [500 level] Consider a model of the Earth’s atmosphere as an ideal gas in equilibrium, at constant temperature in a uniform gravitational field. (a) Derive an expression relating pressure and mass density of the gas. Identify any constants that you may need, but do not do any numerical calculations. (b) Derive an expression for the mass density as a function of height above the surface. (c) Estimate the height above the surface at which the atmosphere is 100 times thinner than at the surface.
- Lagrangian Mechanics [500 level] A smooth horizontal plane has a small hole. A string going through the whole connects two masses, ms and mh, on the surface, and hanging from the string, respectively. The hanging mass is constrained to move only vertically, while the mass on the surface is constrained to move in the plane. The string is
always stretched straight, and the whole system is frictionless.
ms
mh
(a) Find the Lagrangian for this system. (b) Derive Lagrange’s equations of motions. (c) There are two constants of motion in this system. Identify them.
- Hamiltonian Mechanics [700 level] Two masses, m 1 and m 2 , move under their mutual gravitational attraction in a uniform external gravitational field whose acceleration is g. Take as generalized coordinates X, Y , and Z for the center of mass (with Z in the direction of g), the distance r between m 1 and m 2 , and the angles θ and φ which specify the direction of the line from m 1 to m 2. (a) Write the kinetic energy in terms of the generalized coordinates. (b) Give expressions for the six generalized momenta. (c) Give the Hamiltonian for the system. (d) Find Hamilton’s equations of motion. (w) Identify ignorable (i.e. cyclic) coordinates. Argue that the problem can be reduced to two separate one-degree-of-freedom problems. (f) Derive expressions for the six generalized forces, QX , QY ,... , Qφ.
- Lagrangian Mechanics [700 level] As shown in the figure, three massless springs, with spring constant k, are connected in a circle with a mass m between each. The system is constrained to move along the circle’s circumference.
L
L
m
m
(a) Find the Lagrangian and Hamiltonian of this two-particle system in terms of one suitable generalized coordinate. (b) Is H a constant of the motion? (c) Is the energy of the system conserved?
- Thermodynamics [500 level] (a) Show that during the adiabatic expansion of an ideal gas, the temperature and pressure changes are related by
dT T
γ ´ 1 γ
dP P
where the adiabatic index γ “ CP {CV is the ratio of the specific heats. (b) Assume that air undergoes an ideal gas adiabatic expansion as altitude increases. Derive an expression for the variation of temperature with altitude in terms of the average molecular weight of air and other constants.
- Newtonian Mechanics [500 level] A rope of total mass M and length L is suspended vertically with its lower end touching a weighing scale. The rope is released and falls onto the scale. (a) Find x, the length of rope that has landed on the scale by a time t. (b) What is the reading of the scale as a function of x?
- Lagrangian Mechanics [500 level] Consider the pulley system in the figure. The pulley near the ceiling is fixed and massless. The lower pulley is movable with radius R and mass M.
M m
R
(a) Obtain the Lagrangian of the system in terms of a suitable generalized coordinate. (b) Find the equation of motion for this system. (c) For what values of the parameters does the system have an equilibrium?
- Lagrangian Mechanics [700 level] Consider a particle moving in a potential V px, y, zq. (a) Prove that translational invariance of the Lagrangian in a particular direc- tion implies linear momentum conservation in that direction. (b) Prove that rotational invariance implies angular momentum conservation.
- Lagrangian Mechanics [500 level] The Lagrangian for a spherical pendulum with mass M and string length ` is
L “
M pθ^9 q^2
M psin θ φ^9 q^2 M g` cos θ.
(a) Determine the Lagrangian equations of motion. (b) Determine the momenta canonically conjugate to θ and φ in terms of the coordinates and their time derivatives. (c) Construct the Hamiltonian. (d) Determine Hamilton’s equations of motion.
- Thermodynamics [500 level] A gas of photons at temperature T in a vessel of volume V has energy E “ aV T 4 and pressure P “ E{ 3 V. (a) Write down the Maxwell relation for the temperature as a derivative of the energy.
(a) What is the torque imparted to the ball during such a collision, if the height of the bumper is h? (b) Determine h, for an elastic collision with no slipping.
- Newtonian Mechanics [500 level] Consider the two-dimensional force due to the four identical, ideal springs with force constant k, each under a small displacement from the equilibrium position. Each spring has one end in common with the other springs and one end that slides frictionlessly on one of four guide rods arranged in a square as shown.
(a) Show that the force on a particle attached to the common point is
F “ ´ 2 kxˆi ´ 2 kyˆj.
(b) Is this force conservative? Why? (c) Show that the potential function is
U px, yq “ kpx^2 y^2 q constant
(d) Express F and U in terms of polar coordinates ρ and φ.
- Thermodynamics [500 level] A monoatomic ideal gas of N atoms is initially contained in a vessel of volume V and held at temperature T. (a) The gas is allowed to expand freely to double its volume. What are the temperature, pressure, and internal energy after the expansion? (b) Then the gas is compressed adiabatically by a piston, back to its initial volume V. What are the temperature, pressure, and internal energy after the compression? (c) During which of these two processes is entropy generated?
- Statistical Mechanics [700 level] Consider an ideal gas, with known number of particles N , pressure P , tem- perature T , and molecular mass m, contained within a vessel. (a) Write down the volumetric particle density as a function of particle veloc- ity. (b) How many impacts per second per unit area are there on the chamber walls?
- Newtonian Mechanics [500 level] A cylindrical block of wood of mass density ρw, radius R, and height h is partially immersed in a liquid of mass density ρ` and then released, as shown in the figure.
(a) What is the equilibrium height (relative to the top surface of the block) above the water level zeq?
that result from the equations of motion? (c) Now, let Qpkq “
ř` j“´8 Qj^ exppijkaq^ and^ Rpkq “^
ř` j“´8 Rj^ exppijkaq, where the sum on j is over all particles of mass m for Qpkq or over all particles of mass M for Rpkq. Using the above definitions for Qpkq and Rpkq, which are identified as the normal modes of the system, perform the sum over all of the amplitudes in part b above and obtain the equations for Qpkq and Rpkq. (d) Find the normal mode frequencies (i.e. the dispersion relation) ωpkq for the system. Note that ´π{ 2 a ď k ěă π{ 2 a.
- Thermodynamics [700 level] Blackbody radiation can be treated as a macroscopic thermodynamic system. Its energy density is given by
U “
4 V
c
σT 4
where σ is the Stefan-Boltzmann constant. (a) Determine the form of the fundamental energy, free energy, or enthalpy relation whose independent variables are V and T , and obtain expressions for the pressure and specific heat. (Note that the entropy S for the system vanishes at T “ 0.) (b) Starting from the entropy and internal energy of an ideal gas,
S “ N kB
` ln
V
N
4 πm 3 h^2
U
N
˙ 3 { 2 ff+
U “
N kB T, (1)
compute its Helmholtz free energy and the grand canonical potential.
- Small Oscillations [700 level] A thin circular loop of radius R, with mass m distributed uniformly along its circumference, is free to roll along a horizontal surface without slipping. A point particle of mass m is attached to the inside of the loop and is constrained to slide along the inside perimeter of the loop without friction. The system is in a uniform gravitational acceleration g.
(a) Write down the Lagrangian for this system. (b) Find the equations of motion and any possible equilibrium positions for the particle. (c) Which of the equilibrium positions are stable and which are unstable? (You may qualitatively answer this part, if you wish.) (d) Find the frequency of small amplitude oscillations of the particle about all possible positions of stable equilibrium. Consider your results in the limit M " m and discuss whether they are reasonable.
- Relativistic Kinematics [500 level] A ball of putty of mass m travels at speed v towards another ball of putty, also of mass m, which is at rest (but which is free to move) in the lab frame. They collide and stick together forming a new object. (a) Assume that the collision is head on and the speed v of the incoming ball is nonrelativistic, i.e. much smaller than c, the speed of light. Determine the final speed V and mass M of the new nonrelativistic object after the collision. (b) Again assume that the collision is head on, but now assume that the speed v of the incoming ball is relativistic, so that it can not be neglected compared with c. Determine the final speed V and rest mass M of the new relativistic object after the collision. (c) Compare the nonrelativistic and relativistic cases with respect to the con- servation of total kinetic energy and total mass during the collision. For all cases where there is not conservation, explicitly show non-conservation and qualitatively explain why the value is larger or smaller after the collision.
- Central Force Motion [700 level] A particle of mass m moves under the influence of the central potential V prq “ ´kr´^4. (a) Show that the motion occurs in a plane. Hence, use polar coordinates to write the Lagrangian for the system. Determine all constants of the motion. (b) Make a plot of the effective potential for the radial motion of the parti- cle. Give the general condition for a circular orbit. Does the above potential
