GRE Physics Official Exam #3
Window A is a pane of glass 4 millimeters thick, as shown above. Window B is a sandwich consisting of two extremely thin layers of glass separated by an air gap 2 millimeters thick, as shown above. If the thermal conductivities of glass and air ar e 0.8 W/m*C and 0.025 W/m*C, respectively, then the ratio of the heat flow through window A to the heat flow through window B is (A) 2 (B) 4 (C) 8 (D) 16 (E) 32
(0.8/0.025)*(2/4) = 16. Choice (D).
The Lyman alpha spectral line of hydrogen (lambda = 122 nm) differes by 1.8E-12 m in spectra taken at opposite ends of the Sun's equator. What is the speed of a particle on the equator due to the Sun's rotation, in km/s? (A) 0.22 (B) 2.2 (C) 22 (D) 220 (E) 2200
(1.8E-9)(1.22E-7)(9E16) = 2.2. Choice (B). ?
A man of mass m on an initially stationary boat gets off the boat by leaping to the left in an exactly horizontal direction. Immediately after the leap, the boat, of mass M, is observed to be moving to the right at speed v. How much work did the man do during the leap (both on his own body and on the boat)?
(1/2)(M + M^2/m)V^2 -> looks like the kinetic energy with a somewhat reduced mass involved. Choice (D).
A diatomic molecule is initially in the state phi(Theta, phi) = (5Y1,1 + 3Y5,1 + 2Y5,-1)/(38)^(1/2), where Yl,m is a spherical harmonic. If measurements are made of the total angular momentum quantum number l and of the azimuthal angular momentum quantum number m, what is the probability of obtaining the result l = 5? (A) 36/1444 (B) 9/38 (C) 13/38 (D) 5/(38)^(1/2) (E) 34/38
(3^2 + 2^2)/38 = 13/38. Choice (C).
An open-ended U-tube of uniform cross-sectional area contains water (density 1.0 g/cm^3) standing initially 20 cm from the bottom in each arm. An immiscible liquid of density 4.0 g/cm^3 is added one arm until a layer 5 centimeters high forms, as shown in the figure above. What is the ratio h2/h1 of the heights of the liquid in the two arms? (A) 3/1 (B) 5/2 (C) 2/1 (D) 3/2 (E) 1/1
(4.0 g/cm^3)(5 cm)/2 = 10 cm gives us (20 cm)/(10 cm) = 2/1, which is choice (C).
The approximate number of photons in a femtosecond (10^-15 s) pulse of 600 nm wavelength light from a 10-kW peak-power dye laser is (A) 10^3 (B) 10^7 (C) 10^11 (D) 10^15 (E) 10^18
(6E-7)/(10^-15) ~= 10^7. Choice (B).
A plane electromagnetic wave that is a superposition of two independent orthogonal plane waves and can be written as the real part of E = xE1exp(i[kz - wt]) + yE2exp(i[kz - wt + pi]), where k, w, E1, and E2 are real. - If E2 = E1, the tip of the electric field vector will describe a trajectory that, as viewed along the z-axis from positive z and looking the origin is what? - If the plane wave is split and recombined on a screen after the two portions, which are polarized in the x- and y-directions, have traveled an optical path difference of 2*pi/lambda, the observed average intensity will be proportional to (A) E1^2 + E2^2 (B) E1^2 - E2^2 (C) (E1 + E2)^2 (D) (E1 - E2)^2 (E) 0
- We obviously get a non-orthogonal or non-parallel angle, so the choices are reduced to choice (A) and choice (B). The phase shift in the y-component pushes the vector over to the second quadrant, which causes the 135 degree angle to the +x-axis. - We get another Pythagorean difference, which is the difference of the squares given in choice (B).
The period of a physical pendulum is 2*pi*sqrt(l/mgd), where I is the moment of inertia about the pivot point and d is the distance from the pivot to the center of mass. A circular loop hangs from a nail on a barn wall. The mass of the hoop is 3 kg and its radius is 20 cm. If it is displaced slightly by a passing breeze, what is the period of the resulting oscillations? (A) 0.63 s (B) 1.0 s (C) 1.3 s (D) 1.8 s (E) 2.1 s
2*pi*sqrt[(2*d^2*m)/(mgd)] = 2*pi*sqrt[(2*0.2^2)/(9.8 m/s^2)] = 2*pi*sqrt(8E-3) = 2*pi*
A particle of mass m is acted on by a harmonic force with potential energy function V(x) = m(w^2)(x^2)/2 (a one-dimensional simple harmonic oscillator). If there is a wall at x = 0 so that V = infinity for x < 0, then the energy levels are equal to what?
3hw/2, 7hw/2, 11hw/2, ... Similar to the standard quantum oscillator, but different in the sense we are bumping each state up by a difference of double. Choice (D).
A piano tuner who wishes to tune the note D2 corresponding to a frequency of 440.000 hertz. Which harmonic of D2 (counting the fundamental as the first harmonic) will give the lowest number of beats per second, and approximately how many beats will this be when the two notes are tuned properly?
440.000 Hz - 6(73.416 Hz) = 0.5 Hz. The harmonic is 6 and the number of beats is 0.5. Choice (B).
The binding energy of a heavy nucleus is about 7 million eV per nucleon, whereas the binding energy of a medium-weight nucleus is about 8 million eV per nucleon. Therefore, the total kinetic enregy liberated when a heavy nucleus undergoes symmetryic fission is most nearly (A) 1876 MeV (B) 938 MeV (C) 200 MeV (D) 8 MeV (E) 7 MeV
7^2 + 8^2 = 113 times 2 is 226, I don't know. Choice (C).
Suppose that the gravitational force law between two massive objects were F12 = r12GM1M2/r12^(2 + e), where e is a small positive number. Which of the following statements would be FALSE?
A single planet could move in a stationary noncircular elliptical orbit around the Sun. Choice (C).
Consider a particle moving without friction on a rippled surface, as shown above. Gravity acts down in the negative h direction. The elevation h(x) of the surface is given by h(x) = dcos(kx). If the particle starts at x = 0 with a speed v in the x-direction, for what values of v will the particle stay on the surface at all times?
As the value of k and the distance d get larger, then the less capability of the object will have to be on the surface. Choice (D). v </ sqrt[g/(k^2*d)].
The adiabatic expansion of an ideal gas is described by the equation PV^y = C, where y and C are constants. The work done by the has in expanding adiabatically from the state (V,i, P,i) to (V,f, P,f) is equal to (A) PfPi (B) (Pi + Pf)(Vf - Vi)/2 (C) (PfVf - PiVi)/(1 - y) (D) Pi(Vf^(1 + y) - Vi^(1 + y))/(1 + y) (E) Pf(Vf^(1 - y) - Vi^(1 - y))/(1 + y)
As the value of y gets larger, then the more the gas will have to work for expansion. Choice (C).
For an ideal diatomic gas in thermal equilibrium, the ratio of the molar heat capacity at constant volume at very high temperatures to that at very low temperatures is equal to (A) 1 (B) 5/3 (C) 2 (D) 7/3 (E) 3
At high temperatures, the molar heat capacity is given by 7/3, not 5/3 like for substandard temperatures. Choice (D).
A particle with charge q and momentum p is moving in the horizontal plane under the action of a uniform vertical magnetic field of magnitude B. Measurements are made of the particle's trajectory to determine the "sagitta" s and half-chord length l, as shown in the figure above. Which of the following expressions gives the particle's momentum in terms of q, B, s, and l? (Assume s << l.) (A) qBs^2/2l (B) qBs^2/l (C) qBl/s (D) qBl^2/2s (E) qBl^2/8s
By form, choice (D) makes the most sense, which is qBl^2/2s, where as the length, l, gets bigger then the momentum gets bigger, but as the length of s gets larger, then the smaller the angular momentum. Choice (D).
The circuits below consist of two-element combinations of capacitors, diodes, and resistors. Vin represents an AC-voltage with variable frequency. It is desired to build a circuit for which Vout = Vin at high frequencies and Vout = 0 at low frequencies. What is the following circuits will perform this task?
Capacitate and then resist and then we get choice (E), which gives us Vout = Vin due to the capacitor and Vot = 0 with the resistance.
The figure above shows the trajectory of a particle that is deflected as it moves through the uniform electric field between the parallel plates. There is potential difference V and distance d between the plates, and they have length L. The particle (mass m, charge q) has nonrelativistic speed v before it enters the field, and its direction at this time is perpendicular to the field. For small deflections, which of the following expressions is the best approximation to the deflection angle, theta?
Choice (A), lucky guess. Also, the result has unique units, arctan[(L/d)(Vq/mv^2)].
A string consists of two parts attached at x = 0. The right part of the string (x > 0) has mass u,r per unit length and the left part of the string (x < 0) has mass u,l per unit length. The string tension is T. If a wave of unit amplitude travels along the left part of the string, as shown in the figure above, what is the amplitude of the wave that is transmitted to the right part of the string?
Choice (C) makes the most sense because the amplitude of the waves should become 2sqrt(u,l/u,r)/[1 + sqrt(u,l/u,r)]. Choice (C).
A circular loop of radius R rotates with an angular speed w in a uniform magnetic field B, as shown in a uniform magnetic field B, as shown in the figure above. If the EMF E induced in the loop is E0sin(wt), then the angular speed of the loop is (A) ER/B (B) 2*pi*E/R (C) E/(B*pi*R^2) (D) E0^2/(BR^2) (E) tan^-1(E/Bc)
Choice (C) makes the most senses because the circumference looks like it is in the denominator.
Two ions, 1 and 2 at fixed separation, with spin angular momentum operators S1 and S2, have the interaction Hamiltonian H = -JS1*S2, where J > 0. The values of S1 and S2 are fixed at @1(S1 + 1) and S2(S2 + 1), respectively. Which of the following is the energy of the ground state of the system? (A) 0 (B) -JS1S2 (C) -J[S1(S1 + 1) - S2(S2 + 1)] (D) -(J/2)[(S1 + S2)(S1 + S2 + 1) - S1(S1 + 1) - S2(S2 + 1)]
Choice (D). Looks complex and has some combinations.
A collimated laser beam emerging from a commercial He-Ne laser has a diameter of about 1 mm. In order to convert this beam into a well-collimated beam of diameter 10 mm, two convex lenses are to be used. The first lens is of focal length 1.5 cm and is to be mounted at the output of the laser. What is the focal length, f, of the second lens and how far from the first lens should it be placed? f Distance (A) 4.5 cm 6.0 cm (B) 10 cm 10 cm (C) 10 cm 11.5 cm (D) 15 cm 15 cm (E) 15 cm 16.5 cm
Choice (E). Eyeball it.
A particle of mass m moves in the potential shown above. The period of the motion when the particle has energy E is (A) sqrt(k/m) (B) 2*pi*sqrt(m/k) (C) 2*sqrt(2E/mg^2) (D) pi*sqrt(m/k) + 2*sqrt(2E/mg^2) (E) 2*pi*sqrt(m/k) + 4*sqrt(2E/mg^2)
Combining the motion of the particle with the potential V = (1/2)kx^2 and V = mgx, we should get the energy pi*sqrt(m/k) + 2*sqrt(2E/mg^2). Choice (D).
A particle of mass m moves in a one-dimensional potential V(x) = -ax^2 + bx^4, where a and b are positive constants. The angular frequency of small oscillations about the minima of the potential is equal to (A) pi*(a/2b)^0.5 (B) pi*(a/m)^0.5 (C) (a/mb)^0.5 (D) 2(a/m)^0.5 (E) (a/2m)^0.5
Differentiating gives us: dV/dx = 2ax + 4bx^3 = 0 ==> Let b = m, so we get x = 2(a/m)^0.5. Choice (D).
In an n-type semiconductor, which of the following is true of impurity atoms? (A) They accept electrons from the filled valence band into empty energy levels just above the valence band. (B) They accept electrons from the filled valence band into empty energy levels just below the valence band. (C) They donate electrons to the filled valence band from donor levels just below the conduction band. (D) They donate electrons to the filled valence band from donor levels just above the valence band. (E) They donate electrons to the conduction band from filled donor levels just below the conduction band.
Donation of electrons to the conduction band from filled donor levels just below the conduction band is characteristic of an n-type semiconductor. Also, the converse is true for a p-type semiconductor. Choice (E).
The solution to the Schrondinger equation for a particle bound in a 1-dimensional, infinitely deep potential well, indexed by quantum number n, indicates that in the middle of the well the probability density vanishes for (A) the ground state (n = 1) only (B) states of even n (n = 2, 4, ...) (C) states of odd n (n = 1, 3, ...) (D) all states (n = 1, 2, 3, ...) (E) all states except the ground state
Even functions tend to be symmetric and hence would integrate to 0. Choice (B).
The characteristic distance at which quantum gravitational effects are significant, the Planck length can be determined from a suitable combination of the physical constants G, h, and c. Which of the following correctly gives the Planck length?
G = N*m^2/kg^2, h = J*s, and c = m/s. G = m^3/(kg*s^2) , h = kg*m^2/s and c = m/s. Choice (E) with (Gh/c^3)^(1/2) give us a length or more precisely the Planck length.
The ground-state energy of positronium is most nearly equal to (A) -27.2 eV (B) -13.6 eV (C) -6.8 eV (D) -3.4 eV (E) 13.6 eV
GRE Test Makers love positronium, which has a known experimental ground-state energy of -6.8 eV. Choice (C).
What is a bad idea tactic to use as an experimental mapping in a laboratory experiment?
Gain vs. frequency for a low-pass filter with Vo/Vi = 1/w with a linear graph. Choice (D).
In inertial frame S, two events occur at the same instant in time and 3c*min apart in space. In inertial frame S', the same events occur at 5c*minutes apart. What is the time interval between the events in S'? (A) 0 min (B) 2 min (C) 4 min (D) 8 min (E) 16 min
Pythagorean saves the day again. sqrt(5^2 - 3^2) = sqrt(25 - 9) = sqrt(16) = 4, so we have 4 min. Choice (C).
Small-amplitude standing waves of wavelength lambda occur on a string with tension, T, mass per unit length u, and length L. One end of the string is fixed and the other end is attached to a ring of mass M that slides on a frictionless rod, as shown in the figure above. When gravity is neglected, which of the following conditions correctly determines the wavelength? (You might want to consider the limiting cases M --> 0 and M --> infinity.)
I see a tangent, which means as M --> 0, then u/M =infinity and vice-versa. Choice (B).
Two long conductors are arranged as shown above to form overlapping cylinders, each of radius r, whose centers are separated by a distance d. Current of density J flows into the plane of the page along the shaded part of one conductor and an equal current flows out of the plane of the page along the shaded portion of the other, as shown. What are the magnitude and direction of the magnetic field at point A? (A) (u/2*pi)pi*dJ, in the +y-direction (B) (u/2*pi)d^2*J/r, in the +y-direction (C) (u/2*pi)4d^2*J/r, in the -y-direction (D) (u/2*pi(Jr^2/d, in the -y-direction (E) There is no magnetic field at A.
I would say that Choice (A) is the correct answer because it contains pi. Choice (A).
The mean free path for the molecules of a gas is approximately given by 1/(n*sigma), where n is the number density and sigma is the collision cross section. The mean free path for air molecules at room conditions is approximately (A) 10^-4 m (B) 10^-7 m (C) 10^-10 m (D) 10^-13 m (E) 10^-16 m
If the distance was less than 10^-10 m, we will have an angstrom, which is about as small as the nucleus of an atom. Choice (B) is the most reasonable, due to being not too distant but not extremely close to the atoms.
Positronium is the bound state of an electron and a positron. Consider only the states of zero orbital angular momentum (l = 0). The most probable decay product of any such state of positronium with spin zero (singlet) is (A) 0 photons (B) 1 photons (C) 2 photons (D) 3 photons (E) 4 photons
It is 2 photons, don't forget it. Choice (B).
What is the speed of a particle having a momentum 5 MeV/c and a total relativistic energy of 10 MeV? (A) c (B) 0.75c (C) c/sqrt(3) (D) (1/2)c (E) (1/4)c
Let's do some division, [5 MeV/c]/[10 MeV/c^2] = 0.5c, which is choice (D).
A positively charged particle is moving in the xy-plane in a region where there is a non-zero uniform magnetic field B in the +z-direction and a non-zero uniform electric field E in the +y-direction. Which of the following is a possible trajectory for the particle?
Let's go ripple, some nipples with magnetic therapy. Choice (E), the ripply graph.
Which of the following reasons explains why a photon cannot decay to an electron and a positron (y --> e+ + e-) in free space? (A) Linear momentum and energy are not both conserved. (B) Linear momentum and angular momentum are not both conserved. (C) Angular momentum and parity are not both conserved. (D) Parity and strangeness are not both conserved. (E) Charge and lepton number are not both conserved.
Linear momentum is most likely not conserved and the products are bouncing all over the place. Choice (A).
Consider a potential of the form V(x) = 0, x </ a V(x) = V0, a < x < b V(x) = 0, x >/ b as shown in the figure above. Which of the following wave functions is possible for a particle incident from the left with energy E < V0?
Look for the least sinusoidal functions and we get choice (C) with the steep dip in the middle.
A car travels with constant speed on a circular road on level ground. In the diagram above, F,air is the force of air resistance on the car. Which of the other forces shown best represents the horizontal force of the road on the car's tires? (A) F,a (B) F,b (C) F,c (D) F,d (E) F,e
Look into the circle and you will find a compressing force, which is force (B). Choice (B).
Suppose one mole of an ideal gas undergoes the reversible cycle ABCA shown in the P-V diagram above, where AB is an isotherm. The molar heat capacities are C,p at constant pressure and C,v at constant volume. The net heat added to the has during the cycle is equal to
Most complex and has the correct units, which is choice (E). RT,h*ln(V2/V1) - R(T,h - T,c).
Two identical conducting spheres, A and B, carry equal charge. They are initially separated by a distance much larger than their diameters, and the force between them is F. A third conducting sphere, C, is uncharged. Sphere C is first touched to A, then to B, and then removed. As a result, the between A and B is equal to (A) 0 (B) F/16 (C) F/4 (D) 3F/8 (E) F/2.
Moving my balls around, we get a force of 3F/8. Choice (D).
According to the Standard Model of elementary particles, which of the following is NOT a composite object? (A) Muon (B) Pi-meson (C) Neutron (D) Deutron (E) Alpha particle
Muon is a single entity which is single in nature.
In a voltage amplifier, which of the following is NOT usually a result of introducing negative feedback? (A) Increased amplification (B) Increased bandwidth (C) Increased stability (D) Decreased distortion (E) Decreased voltage gain
Negative feedback provides a decrease in power, which is not associated with "increased amplification". Choice (A).
An experimenter needs to heat a small sample to 900 K, but the only available oven has a maximum temperature of 600 K. Could the experimenter heat the sample to 900 K by using a large lens to concentrate the radiation from the oven onto the sample, as shown above? (A) Yes, if the volume of the oven is at least 3/2 the volume of the sample. (B) Yes, if the area of the front of the oven is at least 3/2 the area of the front of the sample. (C) Yes, if the sample is placed at the focal point of the lens. (D) No, because it would violate conservation of energy. (E) No, because it would violate the second law of thermodynamics.
No, we can not do this because the second law of thermodynamics, because heat is oddly absorbed. Choice (E)>
A coaxial cable has the cross section shown in the figure above. The shaded region is insulated. The regions in which r < a and b < r < c are conducting. A uniform DC current density of total current I flows along the inner part of the cable (b < r < c) in the directions shown. The radial dependence of the magnitude of the magnetic field, H, is shown by which of the following?
One straight rise to the top and a double dip or sliding down to the bottom. Choice (B).
Light of wavelength 500 nm is incident on sodium, with work function 2.28 electron volts. What is the maximum kinetic energy of the ejected photoelectrons? (A) 0.03 eV (B) 0.2 eV (C) 0.6 eV (D) 1.3 eV (E) 2.0 eV
Planck's constant is h = 4.136E-15 eV*s. The work function is W = hf - phi = hc/lambda - phi = (4.136E-15 eV*s)(3.0E8 m/s)/(5.0E-7 m) - 2.28 eV = 2.5 eV - 2.3 eV = 0.2 eV. Choice (B).
An atom moving at speed 0.3c emits an electron along the same direction with speed 0.6c in the internal rest frame of the atom. The speed of the electron in the lab frame is equal to (A) 0.25c (B) 0.51c (C) 0.66c (D) 0.76c (E) 0.90c
Pythagorean stuff sqrt(0.6^2 - 0.3^2) = sqrt(0.36 - 0.09) = sqrt(0.27) = 0.51, so we get choice (B), which is 0.51c.
Two identical 1.0 kg blocks of copper metal, one initially at a temperature T1 = 0 degrees C and the other initially at a temperature T2 = 100 degrees C, are enclosed in a perfectly insulating container. The two blocks are initially separated. When the blocks are placed in contact, they come to equilibrium at a final temperature T,f. The amount of heat exchanged between the two blocks in this process is equal to which of the following? (The specific heat of copper metal is equal to 0.1 kcal/(kg*K) (A) 50 kcal (B) 25 kcal (C) 10 kcal (D) 5 kcal (E) 1 kcal
Q = mC*delta(T) = (1.0 kg)[0.1 kcal/(kg*K)](100 K) = 10 kcal. So half due to the thermal equilibrium is 5 kcal. Choice (D).
The infinite xy-plane is a nonconducting surface, with surface charge density sigma, as measured by an observer at rest on the surface. A second observer moves with velocity vx relative to the surface, at height h above it. Which of the following expressions gives the electric field measured by this second observer?
Relativity gives us the electric field as [sigma/2*e]/sqrt(1 - v^2/c^2). Choice (C).
A small particle of mass m and charge -q is placed at a point P and released. If R >> x, the particle will undergo oscillations along the axis of symmetry with an angular frequency that is equal to
Remember, do not derive for this one. Spot the 3 to cube the radius in the denominator and the square root and you get choice (A).
Two capacitors of capacitances 1.0 uF and 2.0 uF are each charged by being connected across a 5.0 V battery. They are disconnected from the battery and then connected to each other with resistive wires so that the plates of opposite charge are connected together. What will be the magnitude of the final voltage across the 2.0 uF capacitor? (A) 0 V (B) 0.6 V (C) 1.7 V (D) 3.3 V (E) 5.0 V
Reverse the capacitances and we get (1/3)(5.0 V) = 1.7 V. Choice (C).
Two small pith balls, each carrying a charge q, are attached to the ends of a light rod of length d, which is suspended from the celing by a thin torsion-free fiber, as shown in the figure above. There is a uniform magnetic field B, pointing straight down, in the cylindrical region of radius R around the fiber. The system is initially at rest. If the magnetic field is turned off, which of the following describes what happens to the system? (A) It rotates with angular momentum qBR^2. (B) It rotates with angular momentum (1/4)qBd^2. (C) It rotates with angular momentum (1/2)qBRd. (D) IT does not rotate because to do so would violate conservation of angular momentum. (E) It does not move because magnetic forces do no work.
Rotates with standard .angular momentum for a cylinder. Choice (A).
A block of mass m sliding down an incline at constant speed is initially at a height h above the ground, as shown in the figure above. The coefficient of kinetic friction between the mass and the incline is u. If the mass continues to slide down the incline at a constant speed, how much energy is dissipated by friction by the time the mass reaches the bottom of the incline? (A) mgh/u (B) mgh (C) umgh/sin(theta) (D) mghsin(theta) (E) 0
Since the block is moving at constant speed, then there should be no change in the kinetic energy, but there should be a change in the potential energy, where the change in potential energy is mgh. Choice (B).
A light source is at the bottom of a pool of water (the index of refraction of water is 1.33). At what minimum angle of incidence will a ray be totally reflected at the surface? (A) 0 (B) 25 (C) 50 (D) 75 (E) 90
Snell's Law: n1sin(1) = n2sin(2) => sin(2) = 1/1.33 ~= 3/4. sin^-1(3/4) = 50 degrees. Choice (C).
The wave function for a particle constrained to move in one dimension is shown in the graph above (phi = 0 for x </ 0 and x >/ 5). What is the probability that the particle would be found between x = 2 and x = 4? (A) 17/64 (B) 25/64 (C) 5/8 (D) sqrt(5/8) (E) 13/16
Squaring the block areas and taking the part over the whole gives us 13/16. Choice (E).
A helium aatom, mass 4u, travels with nonrelativistic speed v normal to the surface of a certain material, makes an elastic collision with an (essentially free) surface atom, and leaves in the opposite direction with speed 0.6v. The atom on the surface must be an atom of (A) hydrogen, mass 1u (B) helium, mass 4u (C) carbon, mass 12u (D) oxygen, mass 16u (E) silicon, mass 28u
Squaring the mass gives us 16u. Choice (D).
Which of the following curves is characteristics of the specific heat C,v of a metal such as lead, tin, or aluminum in the temperature region where it becomes superconducting?
Superconducting materials give us a sharp rise as temperature increases and then there is a deep dip in the graph where the specific heat increases again. Choice (E).
At a given instant of time, a rigid rotator is in the state phi(theta, phi) = sqrt(3/(4*pi))sin(theta)sin(phi), where theta is the polar angle relative to the z-axis and phi is the azimuthal angle. Measurement will find which of the following possible values of the z-component of the angular momentum, L,z? (A) 0 (B) h/2, -h/2 (C) h, -h (D) 2h, -2h (E) h, 0, -h
Take the Planck's constants and we get h, -h. Choice (C).
The half-life of a pi+ meson at rest is 2.5E-8 s. A beam of pi+ mesons is generated at a point 15 meters from a detector. Only 1/2 of the pi+ mesons live to reach the detector. The speed of the pi+ mesons is (A) (1/2)c (B) sqrt(2/5)c (C) 2c/sqrt(5) (D) c (E) 2c
Take the square-rootish answer, which is choice (C).
A bead is constrained to slide on a frictionless rod that is fixed at an angle, theta, with a vertical axis and is rotating with angular frequency w about the axis, as shown above. Taking the distance s along the rod as the variable, the Lagrangian for the bead is equal to what??
The Lagrangian is given by L = T - V = (1/2)m(ds)^2 + (1/2)(wsSin(theta))^2 - mgs*cos(theta). Choice (E).
A black hole is an object whose gravitational field is so strong that even light cannot escape. To what approximate radius would Earth (mass = 5.98E24 kg) have to be compressed in order to become a black hole? (A) 1 nm (B) 1 um (C) 1 cm (D) 100 m (E) 10 km
The Scharwschild radius is given by the formula r = 2GM/c^2 = 2(6.67E-11)(5.98E24)/(9E16) = 1 cm. Choice (C).
If a singly ionized helium atom in an n = 4 state emits a photon of wavelength 470 nm, which of the following gives the approximate final enregy level, E,f, of the atom, and the n value, n,f, of this final state? E,f (eV) n,f (A) -6.0 3 (B) -6.0 2 (C) -14 2 (D) -14 1 (E) -52 1
The approximate final energy of an atom should be given by the n,f being one less than the original n = 4, which is given by choice (A).
A beam of neutral hydrogen atoms in their ground state is moving into the plane of this page nad passes through a region of a strong inhomogeneous magnetic field that is directed upward in the plane of the page. After the beam passes through this field, a detector would find that it has been (A) deflected upward (B) deflected to the right (C) undeviated (D) split vertically into two beams (E) split horizontally into three beams
The beam of neutral atoms gets split into two after being passed through the detector and we get two vertically split beams. Choice (D).
Which of the following atoms has the lowest ionization potential? (A) 2,4He (B) 7,14N (C) 8,16O (D) 18,40Ar (E) 55,133Cs
The bigger the mass, then the lower the ionization energy tends to be within the representative group. So the second lowest first group member Cs should have the lowest ionization potential. Choice (E).
The capacitor shown in Figure 1 above is charged by connecting switch S to contact a. If switch S is thrown to contact b at time t = 0, which of the curves in Figure 2 above represents the magnitude of the current through the resistor R as a function of time? (A) A (B) B (C) C (D) D (E) E
The current depends solely on the resistor R as given in the diagram, so the initial current is V/R, which is given by curve B. Choice (B).
A sphere of mass m is released from rest in a stationary viscous medium. In addition to the gravitational force of magnitude mg, the sphere experiences a retarding force of magnitude bv, where v is the speed of the sphere and b is a constant. Assume that the buoyant force is negligible. Which of the following statements about the sphere is correct? (A) Its kinetic energy decreases due to the retarding force. (B) Its kinetic energy increases to a maximum, then decreases to zero due to the retarding force. (C) Its speed increases to a maximum, then decreases back to a final terminal speed. (D) Its speed increases monotonically, approaching a terminal speed that depends on b but not on m. (E) Its speed increases monotonically, approaching a terminal speed that depends on both b and m.
The differential equation for the motion of a sphere or other rigid body has a speed which can increase monotonically with the retarding force b*v, where the terminal speed depends on both b and m. Choice (E).
The electric potential at a point P, which is located on the axis of symmetry x from the center of the ring, is given by
The electric potential depends solely on the distance between a point on the disk and the point P. So we get an electric potential of Q/[4*pi*e*sqrt(R^2 + x^2)]. Choice (B).
The figure above represents the trace on the screen of a cathode ray oscilloscope. The screen is graduated in centimeters. The spot on the screen moves horizontally with a constant speed of 0.5 cm/ms, and the vertical scale is 2 V/cm. The signal is a superposition of two oscillations. Whichj of the following are most nearly the observed amplitude and frequency of these two oscillations?
The graph is locked between 2.5 V with 83 Hz and a second oscillation of 1.25 V with 500 Hz. Choice (D).
Three equal masses m are rigidly connected to each other by massless rods of length l forming an equilaterial triangle, as hown above. The assembly is to be given an angular velocity w about an axis perpendicular to the triangle. For fixed w, the ratio of the kinetic energy of the assembly for an axis through B compared with that for an axis through A is equal to (A) 3 (B) 2 (C) 1 (D) 1/2 (E) 1/3
The lighter and more pin-wheelish rod without the exterior lengths l have half the inertia than with the exterior lengths l. Choice (B).
The electronic energy levels of atoms of a certain gas are given by E,n = E1*n^2, where n = 1, 2, 3, .... Assume that transitions are allowed between all levels. If one wanted to construct a laser from this gas by pumping the n = 1 --> n = 3 transition, which energy level of levels would have to be metastable? (A) n = 1 only (B) n = 2 only (C) n = 1 and n = 3 only (D) n = 1, n = 2, and n = 3 (E) None
The middle piece would have to be metastable and hence to get this energy level, then we would want to use n = 2 only. Choice (B).
A charged particle, A, moving at a speed much less than c, decelerates uniformly. A second particle, B, has one-half the mass, twice the charge, three times the velocity, and four times the acceleration of particle A. According to classical electrodynamics, the ratio P,B/P,A of the powers radiated is (A) 16 (B) 32 (C) 48 (D) 64 (E) 72
The power formula is (q*a)^2/(6*pi*e*c^3), where the new power P' = [(2q)^2][(4a)^2] = 64P. Choice (D).
A particle of mass m has the wave function phi(x, t) = e^(iwt)[acos(kx) + Bsin(kx)], where a and B are complex constants and w and k are real constants. The probability current density is equal to which of the following? (Note: a* denotes the complex conjugate of a, and [a]^2 = a*a.) (A) 0 (B) hk/m (C) (hk/2m)([a]^2 + [b]^2) (D) (hk/m)([a]^2 - [b]^2) (E) (hk/2mi)[a*b - b*a]
The probability current raises a red flag when it comes to determining the density function. Choice (E).
Consider two horizontal glass plates with a thin film of air between them. For what values of the thickness of the film of air will the film, as seen by reflected light, appear bright if it is illuminated normally from above by blue light of wavelength 488 nm? (A) 0, 122 nm, 244 nm (B) 0, 122 nm, 366 nm (C) 0, 244 nm, 488 nm (D) 122 nm, 244 nm, 366 nm (E) 122 nm, 366 nm, 610 nm
The spacing gives us choice (E) due to the one-half factor.
A sphere of radius R carries charge density proportional to the square of the distance from the center: p = Ar^2, where A is a positive constant. At a distance of R/2 from the center, the magnitude of the electric field is (A) A/4Pie (B) AR^3/40e (C) AR^3/24E (D) AR^3/5e (E) AR^3/3e
The square and the integration (2*2 + 1 = 5) and then dividing by 4, gives us choice (B).
Two real capacitors of equal capacitance (C1 = C2) are shown in the figure above. Initially, while the switch S is open, one of the capacitors is uncharged and the other carries charge Q0. The energy stored in the charged capcitors C1 and C2 carry charges Q1 and Q2, respectively; the voltages across the capacitors are V1 and V2; and the energies stored i nthe capacitors are U1 and U2. Which of the following statements is INCORRECT?
The superposition of energy is non-existent, so U0 = U1 + U2 in general. Choice (E).
The wave function for identical fermions is antisymmetric under particle interchange. Which of the following is a consequence of this property? (A) Pauli exclusion principle (B) Bohr correspondence principle (C) Heisenberg uncertainty principle (D) Bose-Einstein condensation (E) Fermi's golden rule
The wave function is somewhat determined from the Pauli Exclusion Principle due to the dictation of the number of electrons in the nucleus. Choice (A).
When alpha particles are directed onto atoms in a thin metal foil, some make very close collisions with the nuclei of the atoms and are scattered at large angles. If an alpha particle with an initial kinetic energy of 5 MeV happens to be scattered through an angle of 180 degrees, which of the following must have been its distance of closest approach to the scattering nucleus? (ASsume that metal foilk is made of silver, with Z =50.) (A) 1.22 x 50^(1/3) fm (B) 2.9E-14 m (C) 1.0E-12 m (D) 3.0E-8 m (E) 1.7E-7 m
This pretty damn close, choice (B).
In the spectrum of hydrogen, what is the ratio of the longest wavelength in the Lyman series (n,f = 1) to the longest wavelength in the Balmer series (n,f = 2)? (A) 5/27 (B) 1/3 (C) 4/9 (D) 3/2 (E) 3
This question is abundant, I think I saw this question on another test and the answer is similar and exactly choice (A).
A system consists of N weakly interacting subsystems, each with two internal quantum states with energies 0 and e. The internal energy for this system at absolute temperature T is equal to (A) Ne (B) (3/2)NkT (C) Ne*e^(-e/kT) (D) Ne/[e^(e/kT) + 1] (E) Ne/(1 + e^(-e/kT))
This type of question appears on the Physics GRE quite frequently. The equation for the internal energy for this system at absolute temperature T is equal to Ne/[e^(e/kT) + 1]. Choice (D).
As shown above, a ball of mass m, suspended on the end of a wire, is released from height h and collides elastically,m when it is at its lowest point, with a block of mass 2m at rest on a frictionless surface. After the collision, the ball rises to a final height equal to (A) (1/9)h (B) (1/8)h (C) (1/3)h (D) (1/2)h (E) (2/3)h
Total Energy Conservation: E,i = (1/2)mv^2 + mgh = (1/2)(3m)(v/3)^2 + (3m)gh' = E,f. (1/3)mv^2 = (3m)gh' ==> h' = (1/9)h. Choice (A).
A wire is being wound around a rotating wooden cylinder of radius R. One end of the wire is connected to the axis of the cylindre, as shown in the figure above. The cylinder is placed in a uniform magnetic field of magnitude B parallel to its axis and rotates at N revolutions per second. What is the potential difference between the open ends of the wire? (A) 0 (B) 2*piNBR (C) pi*NBR^2 (D) BR^2/N (E) pi*NBR^2
V = NBA [a basketball league] = NB*pi*R^2. Choice (C).
A 3p electron is found in the 3P,3/2 energy level of a hydrogen atom. Which of the following is true about the electron in this state? (A) It is allowed to make an electric dipole transition to the 2S,1/2 level. (B) It is allowed to make an electric dipole transition to the 2P,1/2 level. (C) It has quantum numbers l = 3, j = 3/2, s = 1/2. (D) It has quantum numbers n = 3, j = l, s = 3/2. (E) It has exactly the same energy as it would in the 3D,3/2 level.
We can make a dipole transition to the next level by dropping to the 2S,1/2 level. Choice (A).
A body of mass m with specific heat C at temperature 500 K is brought into contact with an identical body at temperature 100 K, and the two are isolated from their surroundings. The change in entropy of the system is equal to (A) (4/3)mC (B) mCln(9/5) (C) mCln(3) (D) -mCln(5/3) (E) 0
We get positive work and the equation is mCln(3^2/(500/100)) = mCln(9/5). Choice (B).
The operator a = sqrt(mw/2h)(x + ip/mw), when operating on a harmonic energy eigenstaet phi,n with energy E,n, produces another energy eigenstate whose energy is E,n - hw. Which of the following is true> I. a commutes with the Hamiltonian. II. a is a Hermitian operator and therefore an observable. III. The adjoint operator a^t =/ = a. (A) I only (B) II only (C) III only (D) I and II only (E) I and III only
We only have an adjoint operator, where it is choice (C).
Two pendulums are attached tot a massless spring, as shown in the picture. The arms of the pendulums are of identical lengths l, but the pendulum balls have unequal masses m1 and m2. The initial distance between the masses is the equilibrium length of the spring, which has spring constant K. What is the highest normal mode frequency of this system?
We put all of the pieces together and we get the normal mode frequency of the pendulum as sqrt(g/l + K/m1 + K/m2). Choice (D).
A Gaussian wave packet travels through free space. Which of the following statements about the wave packet are correct for all such wave packets? I. The average momentum of the wave packet. II. The width of the wave packet increases with time, as t --> infinity. III. The amplitude of the wave packet remains constant with time. IV. The narrowers the wave packet is in momentum space, the wider it is in coordinate space. (A) I and III only (B) II and IV only (C) I, II, and IV only (D) II, III, and IV only (E) I, II, III, and IV
We want to have dispersive behavior, which is representative of selections II and IV. Choice (B).
When the beta-decay of 60Co nuclei is observed at low temperatures in a magnetic field that aligns the spins of the nuclei, it is found that the electrons are emitted preferentially in a direction opposite to the 60Co spin direction. Which of the following invariances is violated by this decay? (A) Gauge invariance (B) Time invariance (C) Translation invariance (D) Reflection invariance (E) Rotation invariance
When spin is violated, we want to point at the reflection invariance. Choice (D).
Internal conversion is the process whereby an excited nucleus transfers its energy directly to one of the most tightly bound atomic electrons, causing the electron to be ejected from the atom and leaving the atom in an excited state. The most probable process after an internal conversion electron is ejected from an atom with a high atomic number is that the (A) atom returns to its ground state through inelastic collisions with other atoms (B) atom emits one or several x-rays (C) nucleus emits a gamma-ray (D) nucleus emits an electron (E) nucleus emits a positron
When we are shifting to the excited state, we tend to have an atom emit one or several x-rays. Choice (B).
When it is about the same distance from the Sun as is Jupiter, a spacecraft on a mission to the outer planets has a speed that is 1.5 times the speed of Jupiter in its orbit. Which of the following describes the orbit of the spacecraft about the Sun? (A) Spiral (B) Circle (C) ellipse (D) Parabola (E) Hyperbola
When we get more speed and thrusting we get a hyperbola, similar to comet behavior. Choice (E).
A particle of mass m undergoes harmonic oscillation with period T,0. A force f proportional to the speed v of the particle, f = -bv, is introduced. If the particle continues to oscillate, the period with f acting is (A) larger than T,0 (B) smaller than T,0 (C) independent of b (D) dependent linearly on b (E) constantly changing
With the introduction of air resistance, we know that the particle will start to slow down which causes the period to increase. Choice (A).
The curvature of Mars is such that its surface drops a vertical distance of 2.0 meters for every 3600 meters tangent to the surface. In addition, the gravitational acceleration near its surface is 0.4 times that near the surface of Earth. What is the speed of a golf ball would need to orbit Mars near the surface, ignoring the effects of air resistance? (A) 0.9 km/s (B) 1.8 km/s (C) 3.6 km/s (D) 4.5 km/s (E) 5.4 km/s
a = 4.0 m/s^2 and t = 1 s due to gravity. This question is as easy as it can seem, so 3.6 km/1 s = 3.6 km/s. Choice (C).
A particle of unit mass undergoes one-dimensional motion such that its velocity varies according to v(x) = Bx^-n, where B and n are constants and x is the position of the particle. What is the acceleration of the particle as a function of x?
dv/dx = dv/dt*dt/d. Since we have two x's and we get a subtraction by 1, we should have -nB^2*x^(-2n - 1). Choice (A).
The circuit shown above is in a uniform magnetic field that is into the page and is decreasing in magnitude at a rate of 150 T/s. The ammeter reads (A) 0.15 A (B) 0.35 A (C) 0.50 A (D) 0.65 A (E) 0.80 A
i = V/R = (5.0 V)/(10 ohms) = 0.50 A and the EMF provided is E = Flux*A/R = (0.01 m^2)(150 T/s)/(10 ohms) = 0.15 A. So the current overall becomes i - E = 0.50 A - 0.15 A = 0.35 A. Choice (B).
Consider a single-slit diffraction pattern for slit of width d. It is observed that for light of wavelength 400 nm, the angle between the first minimum and the central maximum is 4E-3 radians. The value of d is (A) 1E-5 m (B) 5E-5 m (C) 1E-4 m (D) 2E-4 m (E) 1E-3 m
m*lambda = d*sin(theta) with m = 1 ==> d = lambda/sin(theta) = (4E-7 m)/(4E-3 rad) = 1E-4 m. Choice (C).
The line integral of u = yi - xj + zk around a circle of radius R in the xy-plane with center at the origin is equal to (A) 0 (B) 2*pi*R (C) 2*pi*R^2 (D) pi*R^2/4 (E) 3R^3
u = yi - xj + zk and dr = dxi + dyj + dzk, which gives us u*dr = ydx - xdy + zdz. Since we get zdz as the only nonzero integratable, then we get 2*pi*R^2. Choice (C).
A lump of clay whose rest mass is 4 kg is traveling at 3/5 the speed of light when it collides head-on with an identical lump going the opposite direction at the same speed. If the two lumps stick together and no energy is radiated away, what is the mass of the composite lump? (A) 4 kg (B) 6.4 kg (C) 8 kg (D) 10 kg (E) 13.3 kg
y = [sqrt(1 - (3/5)^2)]^-1 = [sqrt(1 - 9/25)]^-1 = [sqrt(16/25)]^-1 = 5/4. (5/4)(4kg)(2) [due to two lumps of clay] = 10 kg. Choice (D).