3. Tarzan $(M\mathrm{kg})$ tries to cross a river by swinging from a vine as shown in the figure. The vine is $L$ meter long, and his speed at the bottom of the swing is $V\mathrm{m/s}$. And Tarzan knows that the vine has a breaking strength of $S$ Newton. Find the maximum $V$ that Tarzan would not fall in the river.
(A) $[(S - Mg)M / L]^{1 / 2}$
(B) $(SL / M)^{1 / 2}$
(C) $[2(S - Mg)L / M]^{1 / 2}$
(D) $(2SL / M)^{1 / 2}$
(E) $[(S - Mg)L / M]^{1 / 2}$

7. The figure illustrates that a particle of mass m slides without friction inside a hemispherical bowl of radius $R$. If the particle starts from rest with a small displacement from equilibrium, it moves in simple harmonic motion with an oscillating angular frequency $\omega$. Find the corresponding period $T$.(A) $\sqrt{g / R}$ (B) $2\pi \sqrt{g / R}$ (C) $2\pi \sqrt{2R / g}$ (D) $2\pi \sqrt{R / g}$ (E) $2\pi \sqrt{R / 2g}$

8. A large storage tank shown in the figure, open at the top and filled with water, develops a very small hole in its side at a point $y_{1}$ (m) below the water level. The rate of flow from the leak is found to be $W$ (m³/s). Determine the diameter of the hole.(A) $(4W / \pi)^{1 / 2} \times (2gy_{1})^{1 / 4}$ (B) $(4W / \pi)^{1 / 2} \times (2gy_{1})^{-1 / 4}$(C) $(W / \pi)^{1 / 2} \times (2gy_{1})^{-1 / 4}$ (D) $(4W / \pi) \times (2gy_{1})^{-1 / 4}$(E) $(4W / \pi)^{1 / 2} \times (gy_{1})^{-1 / 4}$

9. A particular heat engine has a mechanical power output of $2.50\mathrm{kW}$ and an efficiency of $25.0\%$. The engine expels $8.00\times 10^{3}\mathrm{J}$ of exhaust energy in each cycle. Find the time interval for each cycle.(A) 0.533 s (B) 5.330 s (C) 0.266 s (D) 2.660 s (E) 1.066 s

10. As shown in the figure, A 0.10 mole sample of $\mathrm{H}_{2}$ gas is contained in the left side of the container, which has equal volumes left and right. The right side is evacuated. When the valve is opened, the gas streams into the right side. What is the entropy change of the gas? (hint: $\ln(2) = 0.69314$, $R = 8.314 \, \text{J/mole} \cdot \text{K}$)(A) 5.76 J/K (B) 57.6 J/K (C) 2.88 J/K (D) 0.576 J/K (E) 0.288 J/K

11. When the switch in the circuit is closed at $t = 0$, what is the time constant for the circuit to reach steady state? Each of the three resistors has resistance $R$, and each of the three capacitors has capacitance $C$. The internal resistance of the battery is negligible.(A) RC (B) 9RC (C) RC/9 (D) R/C(E) none of those answers

12. A particle with charge $q1 = -40.0 \, \text{nC}$ is located on the x-axis at $x = 0.0$. A second particle, with charge $q2 = -20.0 \, \text{nC}$, is located on the x axis at $x = 0.5 \, \text{m}$. Is there a point at a finite distance where the electric potential due to these two charges is zero? (A) Yes; it is to the left of $x = 0.0$ along x axis. (B) Yes; it is between $x = 0.0$ and $x = 0.5 \, \text{m}$ along x axis. (C) Yes; it is to the right of $x = 0.5 \, \text{m}$ along x axis. (D) Yes; but not along the x axis (E) No.

13. A parallel-plate capacitor is charged and then disconnected from the battery. The plate separation is then reduced to half its original value. By what factor does the stored energy in the capacitor change? (A) It remains the same. (B) It is doubled. (C) It decreases by a factor of 2. (D) It increases by a factor of 4. (E) It decreases by a factor of 4

14. A charged particle moves through a uniform magnetic field. Which of the following statements about the magnetic force on the particle is true? (A) It exerts a force on the particle parallel to the field. (B) It exerts a force on the particle along the direction of its motion. (C) It increases the kinetic energy of the particle. (D) It changes the momentum of the particle. (E) None of the above.

15. A light ray travels from medium 1 into medium 2, and then into medium 3, as shown in the figure. Which of the following correctly represents the magnitudes of the relative index of refraction, n1, n2, n3? (A) $n3 > n1 > n2$ (B) $n3 > n2 > n1$ (C) $n2 > n3 > n1$ (D) $n1 > n3 > n2$ (E) none of the above. 

16. A plane electromagnetic wave in vacuum propagates in the $+\mathbf{x}$ direction with frequency $f = 900\mathrm{Mhz}$. The magnetic field oscillates perpendicular to $\mathbf{x}$ with amplitude $B_0 = 2.0uT$. What is the expression for electrical field $E(x,t)$? (Use $c = 3\times 10^{8}m / s$). (A) $6.67 \times 10^{-14} \cos(18.85x - 5.65 \times 10^{9}t)$ (B) $600 \cos(18.85x - 5.65 \times 10^{9}t)$ (C) $6.67 \times 10^{-1} \cos(0.33x - 5.65 \times 10^{9}t)$ (D) $600 \cos(0.33x - 5.65 \times 10^{9}t)$

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