gr0877_solutions

GR0877 SOLUTIONS Detailed solutions to the GRE Physics Test by physicsworks (unless otherwise credited) Version 1.1. This solution manual is not officially endorsed by ETS. Solutions are distributed for free. For typos, questions and suggestions, please contact me via physicsworks@ukr.net Educational Testing Service, ETS, Graduate Record Examinations, and GRE are registered trademarks of Educational Testing Service R© October 15, 2011 1. (B) Once the ball has been released, the only acting force on it is a gravitational force (the problem ignores friction). Since gravity has a zero horizontal component, the ball, as viewed from above, moves in a straight line. Thus, one should eliminate all but choices (B) and (D). The car is moving to the right, while the ball is thrown perpendicularly to this direction, so the initial velocity of the ball is directed to the south-east (and preserve this direction in the future). This is choice (B). 2. (D) Horizontal and vertical motions of the object are independent (if we ignore friction). Therefore, no one really needs the initial horizontal component of the ball’s velocity to determine how long it will take to cover some distance in the vertical direction. The initial vertical component of the velocity is zero, so from H = gt2/2 one gets H = (9.8 · 22)/2 = 9.8 · 2 = 19.6 (m). 3. (E) The dissipating power is P = U2/R, where U is the voltage across the resistor R. So, if you double U , the power will quadruple. Comment: you cannot use P = UI, because if you change U the current I will also change. 4. (E) The loop exactly lies on a magnetic field line of the long wire. Thus, at every point of the loop the magnetic field due to the wire is parallel to the direction of the current I2. Fmag = I2 ∫ (dl×B). 5. (A) What else could be? 6. (E) The nth shell can accommodate 2n2 electrons. Thus, the total number of electrons is 2(12 + 22) = 2 · 5 = 10. 7. (C) From mv2/2 = 3kT/2 it’s easy to obtain √ 3kT/m which is choice (C). 8. (D) This is a straightforward application of the Stefan-Boltzmann law. If the temperature is increased by the factor of two, the radiated power and the mass of the ice that can be melt will increase by the factor of 24 = 16. 9. (E) Statements II, I and III represent, respectively, the first, the second and the third Kepler’s law. Thus, choice (E). 10. (B) From the conservation of energy one has ks2/2 = mv2/2 and s = v √ m/k. 11. (C) The energy levels of a quantum harmonic oscillator are En = ~ω ( n+ 1 2 ) , for n = 0, 1, 2, . . . The ground state is the lowest-energy state with n = 0 and E = 1 2 ~ω. 12. (C) One of the Bohr’s postulates is that the angular momentum of the electron is an integer multiple of ~: mvrn = n~. From this, one has mv = n~/rn which is (C). 13. (A) On a log-log plot a function of the form y = axm will appear as a straight line, where m is the slope of the line and a is the y-value corresponding to x = 1. Indeed, log10 y = log10(ax m) = log10 a + m log10 x. Thus, if x = 1, then y = a and m = log10 y2 − log10 y1 log10 x2 − log10 x1 = log10(y2/y1) log10(x2/x1 ). In our case, for x = 1, y ≈ 6 and m = log10(100/10) log10(300/3) = 1 2 . Therefore, y = 6 √ x. 14. (B) (See a wonderful book by John R. Taylor An introduction to error analysis, Chapter 7) If we have N separate measurements of a quantity x: x1 ± σ1, x2 ± σ2, . . . , xN ± σN , then the best estimate is the weighted average xwav = ∑N i=1wixi∑N i=1wi , where wi = 1/σ 2 i , and the uncertainty in xwav is σwav = (∑N i=1wi )−1/2 . Thus, σwav = ( 1 12 + 1 22 )−1/2 = √ 4/5 = 2/ √ 5. 1 15. (E) Remember a general formula for lenses 1 F = (n− 1) ( 1 R1 − 1 R2 ) . Here R1 is the radius of curvature of the lens surface closest to the light source and R2 is the radius of curvature of the lens surface farthest from the light source. Sign convention: the radius of curvature is positive if the center of spherical surface lies to the right from the lens and is negative if the center of the spherical surface lies to the left. Let the source of light be situated to the left from each of the lenses. Then, for (A) one has R1 = −R, R2 = R and FA ∼ −R/2. For (B): R1 = ∞, R2 = R, FB ∼ −R. (C): R1 = ∞, R2 = −R, FC ∼ R. For (D) and (E): R1 = R, R2 = −R, F ∼ R/2. But RE < RD, therefore FE < FD. 16. (D) When unpolarized light passes through the first polarizer it loses one-half of its intensity. Why half? Here is an explanation. A beam of unpolarized light is nothing but a uniform mixture of linear polarizations at all possible angles. When a perfect polarizer is placed in a polarized beam of light, the intensity of the light that passes through is I = I0 cos 2 θ (Malus’ law). The average value of cos2 θ is 1/2, therefore the intensity is half of the initial: Ifirst = I0/2. After the first polarizer the light is linearly polarized and the fraction of light that passes through the second polarizer is (again, according to Malus’ law): I = Ifirst cos 2 45◦ = I0/4 or 25%. 17. (A) Let the Gaussian surface be a cylinder of radius r and length L. The axis of the cylinder coincides with the wire. Gauss’s law: 2piRLE = λL/�0, from which E = 1 2pi�0 λ r . The answer can also be found by elimination of choices, since (A) is the only choice with correct dimensions. 18. (C) This is an application of Lenz’s law. As the magnet enters the loop, the flux through the loop increases. The induced current generates a magnetic field that is opposing the bar magnet’s field. This current is counter-clockwise (that is, form b to a). As the magnet leaves the loop, the flux decreases and the current flows clockwise. Choice (E). 19. (A) According to Wien’s law λ1T1 = λ2T2 = b, from which λ2 = λ1T1/T2 = 500 ·6000/300 = 104 (nm) or 10 µm. 20. (A) The temperature dependence of the radiation is Tr ∝ a−1, where a is a cosmic scale factor. Thus, when the temperature was higher by the factor of 4 (12/3 = 4), the distances were four times less than today. See Misner, Thorne and Wheeler, Gravitation, Chapter 28. 21. (C) For an adiabatic process PV γ = const. If one use PV/T = const for an ideal gas, one has P = const · T/V . Plugging this into PV = const yields TV γ−1 = const. 22. (C) E = γmec 2 ≡ 4mec2. From this, γ = 4 and v = c √ 1− 1/γ2 = √15c/4. The momen- tum: p = γmev = √ 15mec. 23. (B) Imagine the earth (S frame, x-axis is to the right and passes through spaceships) and two spaceships approach to the earth from the left and right with equal speeds v. Let’s move to a reference frame of the left spaceship (system S ′, two systems are oriented the same way). Then, according to the velocity addition rule, velocity of the right spaceship in this frame of reference is: u′r = ur − V 1− urV/c2 , where ur = −v is the velocity of the right spaceship in the S frame and V denotes the speed of S ′ relative to S, V = v. Thus, u′r = −2v 1 + v2/c2 . But this is exactly the speed with which two spaceships approach one another (as seeing from their reference frames). And we know from the length contraction formula that, if the relative speed between two frames is u, a stick at rest with respect to one reference frame is observed from the other reference frame to be contracted by the factor of γ = 1/ √ 1− u2/c2 = l0/l. Problem statement suggests γ = l0/l = 1/0.6 = 5/3. Substituting u ′ r into the equation for γ and working with units in which c = 1: 2 5 3 = 1√ 1− 4v2 (1+v2)2 = 1 + v2 1− v2 , 5− 5v2 = 3 + 3v2, 2 = 8v2, v = 0.5 or v = 0.5c as in choice (B). 24. (B) To the observer, a meter stick l0 that passes through with the speed v = 0.8c is observed (sorry for tautology) to be contracted by the factor of γ = 1/ √ 1− (0.8)2 = 5 3 . Thus, the time it takes the stick to pass the observer is ∆t = l0 γv = 3 5 · 0.8 · 3 · 108 = 2.5 (ns). 25. (E) The only choice which guarantees that the functions are both normalized and mutually orthogonal is (E). 26. (D) The probability that the electron would be found between r and r+dr is P = |ψ|2dV = |ψ|24pir2dr = p(r)dr. The most probable value is given by the maximum of the probability density p(r) (take the second derivative if you want to convince yourself that it is indeed the maximum): dP/dr = 4pir2d|ψ|2/dr + 8pir|ψ|2 = 0. Plugging in: −4pir 2 pia30 2 a0 e−2r/r0 + 8pir pia30 e−2r/r0 = 0. From this, one has r = a0 which is nothing else but the Bohr radius. 27. (C) This is an application of the energy-time uncertainty principle: ∆E∆t & h. Rewriting ∆E as h∆ν one has ∆ν ∼ 1 ∆t = 1 τ = 1 1.6 · 10−9 ≈ 600 (MHz), which is closest to (C). 28. (D) 2(kx2/2) = K(x/2)2/2 gives K = 8k. 29. (C) In elastic collisions energy is conserved. So Mv2/2 = M(v/2)2/2 + Mu2/2 from which one has u = √ 3 2 v. 30. (D) I bet one half of the test takers who did this wrong (according to the official practice book only 51 per cent of all 14,395 examinees who took the Physics Test between July 1, 2007 and June 30, 2010 answered this question correctly) simply mixed up choices (C) and (D). 31. (C) Applying Archimedes’ principle: ρgV = ρwater · g(3V )/4 +ρoil · gV/4 =⇒ ρ = 34ρwater + 1 4 ρoil = 750 + 200 = 950 (kg/m 3). 32. (A) Who says Bernoulli’s principle is unlikely on the PGRE? According to this principle, one has P0 + ρv20 2 = P + ρv 2 2 (there is no ρgz term here because of horizontality of the pipe). Conservation of mass gives: ρv0S = ρvS =⇒ v = 4v0, since S = pir2 = (pir20)/4 = S0/4. Plugging v = 4v0 into the first equation one finally obtain P = P0+ ρv20 2 −16ρv20 2 = P0− 152 ρv20. 33. (E) According to the first law of thermodynamics dS = 1 T dU + P T dV = mcdT T +νRdV V , where c is the specific heat (per one kilogram). Assuming water is incompressible fluid one has dS = mcdT T . Integrating this from T1 to T2 one obtain mc ln T2 T1 . 34. (C) The first law of thermodynamics: Q = 3 2 νR∆T , Q′ = 3 2 νR∆T + p∆V = 3 2 νR∆T + νR∆T = 5 2 νR∆T . Here we have used PV = νRT to rewrite the second term. Thus, Q′ = 5Q/3. 35. (B) Assuming a heat pump is an ideal Carnot engine for its efficiency one has η = W/QH = 1 − TC/TH , where W is the work done by the system, QH is the heat put into the system, TC , TH are the absolute temperatures of the cold and hot reservoirs. W = QH(1−TC/TH) = 15000 · (1− 280/300) = 15000/15 = 1000 (J). 3 36. (A) The magnetic energy is LI2/2. In LC-contour with initial conditions q(t = 0) = q0 and I0 ≡ (dq/dt)0 = 0 the charge on the capacitor is a cosine function of time and the current is, therefore, a sine function. Thus, magnetic energy is a square of the sine function of time with some proportionality factor. That is, it passes through the origin and has a form similar to (A). 37. (E) Clearly, the electric field is in the −x direction, therefore, only (C) and (E) survive. The magnitude of the electric field at P is E = 2Eq cos θ = 2E−q cos θ where Eq = E−q are the magnitudes of the electric field at point P due to the charges q and −q, respectively; θ is the angle between x-axis and the line passing through the point P and charge +q. cos θ = l/2√ r2 + l2/4 =⇒ E = 1 4pi�0 2q r2 + l2/4 l/2√ r2 + l2/4 . For r � l we have E ≈ 1 4pi�0 ql r3 . 38. (E) Magnetic field at distance a from a wire carrying a current I is B = µ0I 2pia . From the picture (and right-hand rule) it is obvious that the magnetic field at point P due to the horizontal wire has the same magnitude, but opposite direction, than of the magnetic field at this point due to the vertical wire. Thus, the net magnetic field is zero. 39. (C) The lifetime of a muon in the laboratory is τl = γτ , where γ = 1/ √ 1− 16/25 = 5/3. The mean distance traveled is l = vτl = vγτ = 4·5 5·3 · 3 · 108 · 2.2 · 10−6 = 880 (m). 40. (B) Conservation of momentum immediately suggests that after the decay particle m and massless particle have the same momentum p. From the relativistic energy-momentum equation (we use units in which c = 1) E2 − p2 = m2 one gets E = p for massless particle. The conservation of energy: M = p+ √ m2 + p2. Moving p to the left-hand side and squaring results in (M−p)2 = m2+p2 or M2−2Mp = m2, from which one has p = (M2−m2)/(2M). Clearly, the decay is possible if M > m. 41. (B) According to Einstein’s photoelectric equation: hν = ϕ + Kmax, where Kmax is the maximum kinetic energy of the ejected electron. Kmax = eV , where V is the stopping potential. From these two equations: V = h e ν − ϕ/e and the slope is h e . 42. (E) From the picture we see that the period of oscillations (this is 2pi phase difference) is approximately 6 cm, while the difference between two points of equal voltage-level (say, zero-level) for two waveforms is 2 cm. Thus, the phase difference between waveforms is 2 · 2pi/6 = 2pi/3. 43. (D) If you took solid state physics course, even at introductory level, you should remember this. 44. (D) The attraction in cooper pair is due to the electron-phonon interaction. 45. (C) According to the general Doppler formula fo = c+ vo c+ vs f , where fo is observed frequency, c is the speed of sound waves in the medium, vs and vo are the speed of the source and the observer relative to the medium, respectively. Sign convention: the speed vs is taken to be positive if the source is moving away from the observer at speed vs, while vo is taken to be positive if the observer is moving toward the source at speed vo. In our case, the source and observer are moving in such a way that vo = vs (and equal to the speed of the wind w). Thus, we have no Doppler effect at all : fo = f . 46. (D) The first minimum is determined by d sin θ = λ or d sin θ = c/ν. Thus, ν = c d sin θ ≈ 350 0.14 · 0.7 = 500 0.14 ≈ 7 · 500 = 3500 (Hz). 4 47. (D) For a pipe of length L, closed at one end and open at the other the resonant frequencies are given by f = v λ = nv 4L , where n = 1, 3, 5, . . . and v is the speed of the sound. Thus, a fundamental frequency is f(n = 1) = v 4L and the next harmonic has a frequency f(n = 3) = 3v 4L . For this next harmonic f(n = 3) = 3f(n = 1) = 3 · 131 = 393 (Hz). 48. (C) We have one NOR-gate with two inverted inputs, one AND-gate and one NAND-gate (you should remember this from your electrical engineering course to solve this problem). For NOR-gate with inverted inputs the output is A+B, for NAND-gate C ·D. Thus, E = A+B · C ·D. 49. (D) The only choice in which free atoms involved is (D). 50. (C) Dimension analysis works fine here. However, you can also derive needed expression using Newton’s second law and Bohr’s quantization rule. Indeed, the former gives mv2/r = Ze2/(4pi�0r 2), while the later mvr = n~. From these two equations (excluding v): rn ∝ n2 me2Z . Potential energy: Epot = −Ze2/(4pi�0r). The total energy: E = Ekin + Epot = −Ze2/(8pi�0r). Substituting rn into the expression for the total energy yields E ∝ mZ 2e4 n2 . To account for the motion of the nuclei we can treat m in the last equation as the reduced mass. Thus, choice (C). 51. (D) I: true; II: false, an atom can emit photons of light only with energy equal to the energy difference between two quantum states; III: true. 52. (C) According to Bragg’s law 2d sin θ = nλ, where θ is the angle between incident ray and scattering planes. In our case n = 1 and d = λ 2 sin θ ≈ 2λ = 0.500 (nm). 53. (D) According to the problem statement, the angular momenta are the same: m1v1R = m2v2R, therefore m1/m2 = v2/v1. For the orbital periods one has T1 = 2piR/v1 and T2 = 2piR/v2 (orbits are circles!). Thus, m1/m2 = v2/v1 = T1/T2 = 3. Think about why cannot we apply Kepler’s third law in its usual form. 54. (E) A solar-mass black hole would exert no more gravitational pull than our sun. Thus, the orbits would remain unchanged. 55. (A) Applying relativistic Doppler effect formula: λ0 λlab = √ 1 + β 1− β ≈ 4 3 . Thus, 16 − 16β = 9 + 9β =⇒ β = 7/25 = 0.28. 56. (D) To fly due north a pilot should point the plane in such a direction that v + u ‖ SN , where v is the velocity of the plain in still air, u is the velocity of the wind, and SN is the south-north line. According to the velocity addition formula, the velocity of the plane in the north direction is V = v + u, V ⊥ u. From this we have (V − u)2 = (v)2, V 2 + u2 = v2, V = √ v2 − u2, t = L/V = L/√v2 − u2 = 500/√2002 − 302 = = 50/√400− 9 = 50/√391 (h). 57. (B) For both figures accelerations of masses 2m and m are the same and can be calculated via Newton’s second law for the whole system of two bodies: 3ma = F , a = F/(3m). For the first figure ma = F12 and F12 = F/3. For the second figure 2ma = F12 and F12 = 2F/3. Thus, choice (B). 5 58. (A) The only force that provides acceleration a for the block B is the friction force due to the block A. Therefore, by Newton’s second law, this force is equal to mBa = 10 kg · 2 m/s2 = 20 N. 59. (C) Elimination of choices. (A): wrong, the period must depends on a; (B): for a particular case a = g the period blows up — wrong; (C): reasonable; (D): for a = 0, T = 0 — wrong; (E): for a = 0, T blows up — wrong. Thus, choice (C). 60. (C) Take a point (x, 0, 0) on the x-axis. At this point the magnetic field due to the wire along zˆ is µ0I/(2pix); due to the wire crossing the first and third quadrants is µ0I 2pix sin 45◦ ; due to the wire crossing the second and fourth quadrants is µ0I 2pix sin 45◦ . Thus, the net magnetic field at point (x, 0, 0) is µ0I 2pix (1 + 2 √ 2). 61. (E) Newton’s second law: mv2/R = qvB or q m · d = const (v and B are constants). If we double q m , the value of d should decrease by the factor of 2. 62. (E) Gauss’s law: Φtotal = ΦA+Φ = q/�0; Φ = q/�0−ΦA = = 1 · 10 −9 8.85 · 10−12 +100 ≈ 100+100 = 200 (N ·m2/C). Check the table of information in your test book for the numeric value of �0 (if you don’t remember). 63. (D) Given example is a β+ decay. Beta decay is mediated by the weak force. 64. (D) Eigenvalues of L2 are ~2l(l+1), eigenvalues of Lz are m~. Thus, ( √ 2~)2 = ~2l(l+1), from which l = 1. For a given l, there are 2l+ 1 different values of m: m = −l,−l+ 1, . . . , l− 1, l. In our case, m = −1, 0, 1 and, therefore, the possible values of Lz are −~, 0, ~. 65. (C) Step-by-step elimination. I: true: En = ~ω(n + 1/2), this is indeed evenly spaced spectrum; II: false, the potential function is a quadratic function, not linear; III: false, the expectation value of both potential and kinetic energy is half the total energy 〈V 〉 = 〈T 〉 = 1 2 ~ω(1 + 1/2). Thus, for n = 0, 〈T 〉 6= 0. IV: true. 66. (D) To account for the fact that the nucleus has non-infinite mass and moves around atom’s CM one must replace the electron mass (muon mass for the muonic atom) with the reduced mass µ. Noting that Rydberg R is proportional to µ we conclude that if we replace electron me with the muon mµ the R is changed by the factor of µmuon µelectron = mpmµ mp +mµ mp +me mpme . 67. (D) At any instant of time an electric field inside the parallel-plate capacitor is E = σ �0 = Q �0S , where σ is a surface density of a charge Q on a positive plate. Differentiating this with respect to t gives dE dt = dQ/dt �0S = I �0S = 9 8.85 · 10−12 · (0.5)2 ≈ 1012 0.25 = 4 · 1012 ( V m · s ) . 68. (D) The total resistance of the circuit is equal to the resistance of three R+R = 2R resistors connected in parallel: 1 Rtotal = 1 2R + 1 2R + 1 2R = 3 2R , Rtotal = 2R 3 . The current flowing through the battery is I = V Rtotal = 3V 2R . 69. (D) The impedance of an ideal resistor is ZR = R, and the impedance of an ideal capacitor is ZC = 1 iωC , where i is imaginary unit. The total impedance of the circuit is Z = ZR +ZC = 6 R+ 1 iωC . The total current is I = Vi/Z = Vi R + 1 iωC . The amplitude Vo of the output voltage is Vo = IZC = Vi( R + 1 iωC ) iωC = Vi iRCω + 1 . From this one has G = Vo Vi = 1 iRCω + 1 . Therefore, when ω → ∞, G → 0 and when ω → 0, G → 1. The only graph with such a behavior of G