Sample Problems inClassical Mechanics

1 Sample Problems in Classical Mechanics1. Two particles move about each other in circular orbits under the influence of mutualgravitational force, with a period . At some timet= 0, they are suddenly stoppedand then they are released and allowed to fall into each other. Find the timeTafterwhich they collide, in terms of .2. Solve either (a) below or assuming (a) solve (b).(a) Show that for any repulsive central force, a formal solution for the scattering angle is given by, = + 2 u00sdu 1 V(u)=E s2u2whereu= 1=r,Vis the potential energy,Eis the total energy,sis the impact parameterandu0corresponds to the turning point of the orbit.(b) Now consider a repulsive central forcef=kr 3. Show that ( )d =k2E(1 x)dxx2(2 x)2sin( x)wherex = andE >0 is the Consider central force motion with the force given by,f(r) = Kr2+Cr3 Obtain the orbit equation and show that the orbit is a precessing ellipse.

2 What is theapproximate rate of precession to first order inC?4. On a 2N dimensional phase space, consider the set of functions which are purelyquadratic in the generalised coordinates and momenta.(a) Show that the Poisson bracket of any two members of the setis again in the set.(b) Let the Hamiltonian be given by,H=12N i=1{(pi)2+ (qi)2}Which functions in the above set constitute infinitesimal symmetries of the sys-tem? How many independent symmetries are there?15. Consider a one dimensional system with Hamiltonian given by,H(q; p) =12[ap2+bq2+ 2cpq]wherea; b; care a canonical transformation (q; p) (Q; P) such thatH(Q; P) =AP2+BQ2:What is the frequency of the harmonic oscillator in the new variables? Is it the sameas in the old variables?6. Consider Newton s cradle in the figure below.

3 It is well known that ifkballs are liftedtogether and released, after collision, exactlykballs move out at the other end. Thetask is to explain why. To simplify the problem , first consider a cradle with two ballsofun-equal masses. The 1st ball with massm1is used to strike the stationary one ofmassm2. Is the state ( the two momenta) after the collision unique? If so, give the23451final momenta of the two balls as fractions of the initial momentum. What happenswhen the two masses are equal?Now consider the case of three balls ofequalmasses. As before, one ball at one extremeis used to strike. Is the final state unique? If not, why does one see only one ball exitthe group, at the other extreme?Comment on generalization toN(not too large) balls in the : Sample questions for Comprehensive ExamsComments* The questions below (except4) are from Jackson, with a few changeshere and there.

4 * I did not write down the questions for many topics, notably radiatingsystems.* There are many internet sites with solutions to (most of) Jackson sexercises. Anyway, I ll attach the solutions later either downloading fromsome site, or my own. But there are often more than one way of solving agiven problem , including the ones Questions(1)Consider an electrostatic potential given by =q4 0e arr(1 +br)whereaandbare constants. Find the distribution of charge (both continuousand discrete) that will give this potential.(a)What, if anything, is special whena= 2b?(a)Interpret your results physically. (2)A point chargeqis at a distancedaway from an infinite plane con-ductor held at zero potential. Using the method of images (or otherwise),find:1(a)surface charge density induced on the plane, and plot it.

5 (b)the force between the plane and the charge(c)the work necessary to remove the chargeqfrom its position to infinity(d)express the answer in part(c)in electron volts whend= 1 Angstrom. (3)A straight-line charge with linear charge density is located perpen-dicular to thex yplane at (x0; y0) in the first quadrant. The intersectingplanesx= 0; y 0 andy= 0; x 0 are conducting boundary surfaces heldat zero potential. Consider the potential and fields in the first quadrant.(a)The electrostatic potential for an isolated line charge at (x0; y0) isgiven by (x; y) = 4 0lnR2r2whereRis a constant andr2= (x x0)2+ (y y0)2. Find the potential for the line charge in the presence of the intersecting planes. Verify explicitlythat the potential and the tangential electric field vanish on the bundarysurfaces.

6 (b)Show that far from the origin, the potential asymptotes to 4 0(x0y0) (xy) 4where 2=x2+y2 (x0)2+ (y0)2.** If you are unable to solve this problem , then solve the simpler oneby assuming ONLY ONE conducting plane. The expression in part(b)willnow be different. The correct solution will be given 60 percent marks. ** (4)Consider three charges: a charge ( 2q) is placed at (0;0;0); a charge2(+q) is placed at (0;0; a); and, the third charge (+q) is placed at (0;0; a).Find the point(s), if any, at finite distance(s) where the netforce points are stationary points. Infinitesimal motion away from suchpoints in any of the three independent directions may be stable or unstable insome or all directions. The possibilities are: (1) stable inall three directions;(2) stable in two and unstable in the other one direction; (3)stable in oneand unstable in the other two directions; (4) unstable in allthree directions.

7 (a)Which of the possibilities are realised by the stationary point(s) ofthe three charge distribution given above.(b)Note that, for static charge distributions, not all the abovefour pos-sibilities may be realised. If you assert that some of the possibilities cannotbe realised by any static charge distributions then: which possibities or they?Prove your assertion.(c)By giving atleast one example of charge distribution (besides the ex-ample given above), show that each of the remaining possibilities is realised.(That is, in your examples, you must find all the stationary points and anal-yse explicitly the infinitesimal motions near them.) (5)A transverse plane wave is incident normally in vacuum on a perfectlyabsorbing flat screen.(a)From the law of conservation of linear momentum, show that thepressure radiation pressure exerted on the screen is equal to the fieldenergy per unit volume in the wave.

8 (b)Let the incident radiation have a flux of 1:4kW=m2. The absorbingscreen has a mass of 1gm=m2. What is the screen s acceleration due toradiation pressure? 3(6)Consider a layered slab of thicknessd, with its bottom atz= 0 andits top atz=d. The refractive index of the slab isn. The refractive indexof the medium below it, namely forz <0, isn1, and of the medium above,namely forz >0, plane electromagnetic wave is incident normally on the slab from below,namely from thez <0 side.(a)Calculate the reflection and transmission coefficients (ratios of re-flected and transmitted Poynting s flux to the incident flux).(b)Taken1= 3,n= 2, andn2= 1 . For these values, plot the reflectioncoefficient as a function of frequency. (7)Consider a conductor whose conductivity ( ) is frequency dependentand is given by ( ) = 01 i where 0= 0 2p and is damping time.

9 Consider electric fields in a conduc-tor, Ohm s law, continuity equation, and the differential form of Coulomb slaw.(a)Show that the time-Fourier transformed charge density satisfies theequation( ( ) i 0) (~x; ) = 0(b)Assume that p 1 . Show that, in this approximation, an initialdisturbance will oscillate with a frequency pand its amplitude will decaywith a decay constant12 . (8)A particle with massmand chargeqmoves in a uniform static electricfield~E0. Its initial velicity is~v0and is perpendicular to the electric that the particle motion may be (a)Solve for the velocity and position and obtain them explicitly asfunctions of time. Plot these functions.(b)Eliminate time and obtain velocity as a function of the position. Plotthis function.(c)Discuss the motion for short and long times.

10 (9)Static uniform electric and magnetic fields,~Eand~B, make an angle with respect to each other.(a)By a suitable choice of axes, solve the force equation for themotionof a particle of chargeqand massmin rectangular coordinates.(b)For~Eand~Bparallel, show that with appropriate constants of inte-gration et cetera, the parametric solution can be written asx=AR sin ; y=AR cos ;z=R 1 +A2 Cosh( ); ct=R 1 +A2 Sinh( )whereR=mc2qB =EB,Ais an arbitrary constant, and is a parameter(actually,cRtimes the proper time). 5 Sample Questions in Quantum Mechanics1. LetHbe the Hamiltonian of a physical system. Denote by| nitheeigenvectors ofHwith eigenvaluesEn:H| ni=En| ni(a) For an arbitrary operatorAprove the relation:h n|[A,H]| ni= 0(b) Consider a one-dimensional problem , where the physical systemis aparticle of massmand of potential energyV(X).