14. A particle is trapped in a potential well with the depth of E\(_0\). If the wavefunction of the particle is \(\psi(x,t) = A\sin(\omega t + kx) + iA\cos(\omega t + kx)\), where A, \(\omega\) and k are constants and i is the unit imaginary number, what is the probability, per unit length of the x axis, of finding the particle near the coordinate x at time t?
(A)E₀
(B)A²
(C)
(D)A
(E)None of the above

15. Two metal plates are connected to a battery and a current meter. When a light beam with tunable energy E is shined on one of the plates, the values read by the current meter is plotted as a function of E. E0 corresponds to the photon density of 1012cm-3and the photon wavelength of 200 nm. When the incident photon wavelength is increased to 400 nm, what is the photon density to obtain the current value of 5 μA?(A)2✖1012 cm-3(B)4✖1012 cm-3(C)8✖1022cm-3(D)1012-3(E)None of the above

16. Train A travels at 0.8c and passes a station platform. An observer on the platform records a time interval of 2 μs as the front and the rear of the train pass the observer. What is the proper length of the train?(A)288 m(B)320 m(C)480 m(D)600 m(E)None of the above

17. A container contains 2 moles of ideal monatomic gas with volume V0 and temperature T0. What is the change in entropy of the ideal gas if the container is compressed isothermally as its volume is reduced to exactly one half of the initial volume?(A)3R ln 2(B)-3Rln2(C)2Rln2(D)-2Rln2(E)None of the above

18. A 200 g ball is thrown at 30 m/s. It is struck by a bat which gives it a velocity of 40 m/s in the opposite direction. If the time of contact is 10-2s, what is the average force on the ball?(A)1000 N(B)1200 N(C)1400 N(D)1600 N(E)None of the above

19. A quarter circle with a linear density \(\lambda\) kg/m and a radius R as shown in the figure. What is the distance from the center of the quarter circle O to the center of mass of the quarter circle?(A)(B)0(C)(D)(E)None of the above

20. A particle of mass m1 = 2kg moving at velocity u (m/s) makes a one-dimensional elastic collision with a particle of mass m2 at rest. What is the possible mass of m2if it has one-third the initial kinetic energy of m1 after collision?(A)(B)(C)(D)(E)None of the above

21. A marble of radius r rolls without slipping down an incline and then up along a vertical circular track of radius R, as shown in the figure. What is the minimum height H from which the ball must start so that it barely stays in contact at the top of the circle? Assume r ＜＜ H and r ＜＜ R.(A)2 R(B)2.3 R(C)2.7 R(D)3 R(E)None of the above

22. Water emerges at speed \(v_0\) from the opening of a faucet of radius R. In laminar flow, the cross-sectional area of the vertical stream of water decreases as it falls. What is the relation between the radius r of the stream and the vertical drop y?(A)(B)(C)(D)(E)None of the above

23. What is the period T of the spring-block system for two springs with spring constants \(k_1\) and \(k_2\) respectively connected in series as shown in the figure? The mass of the block is m.(A)(B)(C)(D)(E)None of the above

24. A string of length L and linear mass density \(\lambda\) is under a tension F and vibrates in its fundamental mode at f1. What is the new fundamental frequency if the length L increases by 25%, the linear mass densityλdecreases by 20%, and the tension decreases by 20%?(A)1.2 f1(B)0.8 f1(C)0.83f1(D)1.25 f1(E)None of the above

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