40. Consider the continuous function f(x) = 3 cos 2x + 4 sin 2x, and let g(x) = a cos x + b sin x be a linear combination of the functions cos x and sin x.
Define inner product of two functions as (f, g) = f(x)g(x)dx. Please find a and b such that ||f - g||² = |f(x) – g(x)|² dx is minimized. Which of the following shows the correct answer?
(A) a = 4 and b = 3
(B) a = 3 and b = 4
(C) a = 0 and b = 0
(D) a = 3 and b = -4

(E) a = -3 and b = 4

1. A bead of mass m is constrained to move on a massless circular hoop of radius R. The hoop is rotating around its center that is aligned in the z-axis (the direction of gravity g) with a constant angular velocity Ω, see figure below. If the hoop is rotating slowly, this system has a stable equilibrium angular velocity of the hoop exceeds a critical 0 becomes unstable. What is the critical angular velocity? (A)(B) (C) (D) (E) None of the above

2. Two masses m₁ and m₂ placed on a flat surface are connected by a spring of a spring constant k aligning in the x direction; see figure below. Initially, the spring is at its natural length (no compression/extension), and m₁ is at rest while m₂ moves with a velocity in the +x direction. Since there is friction, these masses will eventually stop moving. Assume these two masses are identical, m₁ = m₂ = m, and the gravitational strength is g. What is the minimum friction coefficient required so that m₁ never moves during the slowdown motion of m₂? For simplicity, assume the kinetic and static friction coefficients are the same.(A) (B) (C) (D) (E) None of the above

3. A pendulum is composed of a bob of mass m2, a rigid string of length and negligible mass, and a mass m₁ at the point of support, as shown in the figure. The mass m₁ can only move horizontally in the x-direction on a frictionless track, while m2 swings in the xy plane. The system is placed in a uniform gravitational field =. What is the period of the oscillation when m2 is released at a small angle?(A) (B) (C) (D) (E)

4. Jack is pushing a box moving at a constant speed on a horizontal ground. Due to friction between the box and the ground, Jack has to continue to do positive work to keep the box moving, and the work output per unit time is P. If Jack would like to push the box along a slope of an inclined angle of 18 degrees (relative to the horizontal ground) at the same constant speed, make an estimate of the power output required. Assume the coefficient of the kinetic friction remains the same and equals unity.(A) P(B) 1.26P(C) 1.33P(D) 1.42P(E) 1.51P

5. The membrane serves as an insulator that separates the environments inside and outside the cell. The membrane potential is the difference in electric potential between the interior and the exterior of a biological cell. Assume that the interior of a cell has a lower electric potential compared to the exterior of a cell, and the K+ (potassium ion) concentration inside the cell is higher than that outside the cell. If no ions (such as Na+, Ca2+, etc.) can enter or leave the cell except the potassium ions through some holes on the membrane, which of the following is true?(A) The potassium ions will enter the cell, which increases the membrane potential(B) The potassium ions will leave the cell, which increases the membrane potential(C) The potassium ions will enter the cell, which decreases the membrane potential(D) The potassium ions will leave the cell, which decreases the membrane potential(E) None of the above

6. In the double-slit experiments of neutrons, which of the following statements is true?(A) Each neutron simultaneously passes through two slits and interferes with itself to form an interference pattern(B) Different neutrons pass through two slits and interfere with others to form an interference pattern(C) If one tracks the motion of each neutron (by shining a light), the interference pattern becomes more pronounced(D) It will not form the interference pattern because neutrons, unlike electrons, do not carry charges(E) None of the above

7. An ideal LC circuit with inductor L and capacitor C is initially charged with zero on the capacitor and a maximum instantaneous current of Io. Which of the following is true?(A) The total energy of the LC circuit oscillates with time(B) If one increases the capacitance of the capacitor while keeping the inductance and the initial charges and currents the same, the total energy will not change(C) If one increases the inductance of the inductor while keeping the capacitance and the initial charges and currents the same, the total energy will not change(D) The current will monotonically decay over time and reach zero(E) None of the above

8. For an RL circuit used as a low-pass or high-pass filter, which of the following statements is true?(A) The inductor blocks high-frequency signals since its reactance increases with the frequency(B) The inductor blocks low-frequency signals since its reactance increases with the frequency(C) The inductor blocks high-frequency signals since its reactance decreases with the frequency(D) The inductor blocks low-frequency signals since its reactance decreases with the frequency(E) None of the above

9. A ball of mass m is connected by a string of length L to a pivot fixed in place. A. The ball is originally at rest. A force F applied horizontally to the right on the ball. The force barely balances the gravity and the string tension so that the ball has negligible kinetic energy when it slowly swings up. The string finally makes an angle θ with the vertical line. What is the work done by the force F during this process?(A) mgL(sinθ)(B) mgL(1-cosθ)(C) mgL(cosθ-1)(D) mgL(cosθ)(E) 0

10. Two uniform ladders, both of length L, are hinged at point A and tied together by a horizontal rope at its center. The angle between the ladders is 90 degree. The ladders have mass m₁ and m₂ respectively. Assume the floor is frictionless. Find the tension in the rope.(A)(B) (C) (D) 2m₁g + 2m₂g(E) 0

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