I. PART 1
QUESTION 01
A car at rest starts moving along a straight line and stops at time t = 80 s. The velocity v of the car changes as a function of time t as shown in Fig below. Find the distance that the car travels.
(A) 5000 m
(B) 4000 m
(C) 3000 m
(D) 2000 m
(E) 1000 m
(F) 500 m
QUESTION 02
A small ball at rest falls down from a height of h above the ground and bounces repeatedly. The coefficient of restitution between the ball and the ground is denoted as e, and the acceleration of gravity as g. Find the maximum height of the ball between the nth and the (n+1)th impact with the ground.
(A) \(he\)
(B) \(h(1 - e)^n\)
(C) \(he^n\)
(D) \(he^{2n}\)
(E) \(he^{2n+2}\)
(F) \(he^{2n-2}\)
QUESTION 03
Two long straight wires with the same current I flowing in the opposite direction, are placed parallel to each other with a distance of 2d as shown in Fig. below. Find the magnitude of magnetic field H at point P.
(A) \(\dfrac {I}{2 \pi d}\)
(B) \(\dfrac {I}{\pi d}\)
(C) \(\dfrac {\pi I}{d}\)
(D) \(\dfrac {2 \pi I}{d}\)
(E) \(\dfrac {I}{d}\)
(F) \(\dfrac {I}{2d}\)
QUESTION 04
A sinusoidal wave travels in the positive x-direction with a constant speed of 2 m/s. Figure below shows a snapshot of the wave at t = 0 s as a function of x. Find the formula for the displacement y at time t.
(A) \(2 \sin \pi (x - 2t)\)
(B) \(3 \sin \dfrac {\pi}{2} (x - 2t)\)
(C) \(3 \sin \dfrac {\pi}{4} (x - 2t)\)
(D) \(3 \sin \dfrac {\pi}{2} (2x - t)\)
(E) \(3 \sin \pi (x - t)\)
(F) \(3 \sin \dfrac {\pi}{4} (2x - t)\)
QUESTION 05
A movable piston is fitted in a tube as shown in Fig. below. A speaker near the open end of the tube emits sound waves with a frequency of 555 Hz. When the piston moves from the left end to the right, the first and second resonances are produced at distances 14 cm and 44 cm from the open end, respectively. Find the speed of the sound waves.
(A) 344 m/s
(B) 338 m/s
(C) 333 m/s
(D) 328 m/s
(E) 322 m/s
(F) 311 m/s
II. PART 2
Consider the circuit shown in figure consisting of a resistor R, a capacitor C, an inductor (coil) L, a switch S, and a battery with voltage V. At the beginning, the switch is in position b and the capacitor is uncharged. Then the switch is changed to position a. After having been in position a for a long time, the switch is changed to position c.
QUESTION 01
Find the current \(I_o\) through the resistor just after the switch is changed to position a.
(A) \(CV\)
(B) \(\dfrac 12 CV\)
(C) \(\dfrac {V}{3R}\)
(D) \(\dfrac {V}{2R}\)
(E) \(\dfrac VR\)
(F) 0
QUESTION 02
Find the charge on the capacitor after having been in position a for a long time.
(A) \(C(V + RI_o)\)
(B) \(C(V - RI_o)\)
(C) \(\dfrac 12 CV^2\)
(D) \(CV^2\)
(E) \(\dfrac 12 CV\)
(F) \(CV\)
QUESTION 03
Find the Joule heat generated in the resistor by the time the capacitor is fully charged.
(A) \(0\)
(B) \(\dfrac 12 CV^2\)
(C) \(\dfrac 13 CV^2\)
(D) \(\dfrac 14 CV^2\)
(E) \(\dfrac 23 CV^2\)
(F) \(CV^2\)
QUESTION 04
At time t = 0, the switch is moved from position a to position c. How does the current \(I_L\), which flows into the inductor (coil), change as a function of time? Find one appropriate graph in Fig. below.
QUESTION 05
Find the maximum value of \(I_L\)
(A) \(\sqrt{\dfrac CL} V\)
(B) \(\sqrt{\dfrac LC} V\)
(C) \(\sqrt{LC} V\)
(D) \(\dfrac {1}{\sqrt{LC}} V\)
(E) \(\dfrac VR\)
(F) \(\dfrac {V}{2R}\)
III. PART 3
An object of mass m is moving with a speed v on a frictionless plane as shown in Fig. below. The object reaches at the end of the plane, x = 0 and y = h, at time t = 0, and jumps into the air. There is a slope in the x > 0 region as shown in Fig. The acceleration of gravity is denoted as g.
QUESTION 01
Find the x coordinate of the object at time t (> 0) when the object is in the air.
(A) \(\dfrac 12 vt\)
(B) \(\dfrac 13 vt\)
(C) \(2 vt\)
(D) \(vt\)
(E) \(\dfrac 12 vt^2\)
(F) \(\dfrac {v}{t}\)
QUESTION 02
Find they coordinate of the object at time t (> 0) when the object is in the air.
(A) \(-\dfrac 12 gt^2\)
(B) \(gt^2\)
(C) \(h\)
(D) \(h - gt\)
(E) \(h - gt\)
(F) \(h - \dfrac 12 gt^2\)
QUESTION 03
If v is larger than a speed Ve, the object does not hit the slope and directly drops to the horizontal plane at y = 0. Find the expression of Ve.
(A) \(\dfrac {\sqrt{2gh}}{\tan \theta}\)
(B) \(\dfrac {\sqrt{2gh}}{2}\)
(C) \(\dfrac {\sqrt{2gh}}{2 \tan \theta}\)
(D) \(\sqrt{gh}\)
(E) \(\dfrac {\sqrt{gh}}{2}\)
(F) \(\dfrac {\sqrt{gh}}{\tan \theta}\)
QUESTION 04
If v is less than Ve, the object hits the slope. Find the x coordinate of the impact point.
(A) \(\dfrac {v^2}{g} \sin \theta\)
(B) \(\dfrac {v^2}{g} \tan \theta\)
(C) \(\dfrac {v^2}{g} \cos \theta\)
(D) \(\dfrac {2 v^2}{g} \sin \theta\)
(E) \(\dfrac {2 v^2}{g} \tan \theta\)
(F) \(\dfrac {2 v^2}{g} \cos \theta\)
IV. PART 4
A non-adiabatic cylinder with cross-sectional area \(A\) is placed at normal air pressure and is filled with an ideal gas as shown in Fig. (a). The cylinder is closed by an adiabatic piston with mass \(m\). The pressure of the air is \(P\) and the temperature of the air is \(T\). In equilibrium, the height of the piston is \(h\) from the bottom and the temperature of the gas is \(T\). An object of mass \(M\) is placed on the piston, and the height of the piston becomes \(h_1\) in equilibrium as shown in Fig. (b). The acceleration of gravity is denoted as \(g\).
QUESTION 01
Find the expression of \(h_1\).
(A) \(h\)
(B) \(\dfrac {PA + mg}{PA + (M + m)g} h\)
(C) \(\dfrac {mg}{PA + (M + m)g} h\)
(D) \(\dfrac {PA + (M + m)g}{PA + Mg} h\)
QUESTION 02
Find the work done on the gas by the air when the height of the piston changes from \(h\) to \(h_1\).
(A) 0
(B) \(\dfrac {Mg(PA + Mg)h}{PA + Mg}\)
(C) \(\dfrac {MgPAh}{PA}\)
(D) \(\dfrac {MgPAh}{PA + Mg}\)
(E) \(\dfrac {MgPAh}{PA + mg}\)
(F) \(\dfrac {MgPAh}{PA + (M + m)g}\)
QUESTION 03
We then cover the system with adiabatic walls and remove the object with mass \(M\). In equilibrium, the height of the piston becomes \(h_2\). Find the temperature of the gas.
(A) \(T\)
(B) \(\dfrac {h}{h_2} T\)
(C) \(\dfrac {h_2 + h}{h} T\)
(D) \(\dfrac {h}{h + h_2} T\)
(E) \(\dfrac {h_2}{h} T\)
(F) \(\dfrac {h_2 + h}{h_2} T\)
QUESTION 04
We then remove the adiabatic walls. In equilibrium, the height of the piston becomes \(h_3\). Find the expression of \(h_3\).
(A) \(h\)
(B) \(h_2\)
(C) \(2h\)
(D) \(2h_2\)
(E) \(\dfrac h2\)
(F) \(\dfrac {h_2}{2}\)
V. PART 5
A light ray of wavelength \(\lambda\) traveling through air is incident on a flat thin soap film at an angle \(\theta\) to the normal, as shown in figure below. The thickness of the film is \(d\). The path of the light ray is bent toward the normal in the film with the refraction angle \(\phi\). We assume that the index of refraction of air is 1 and denote the index of refraction of the film by \(n\).
QUESTION 01
Find the wavelength of the light ray in the film.
(A) \(\dfrac {\lambda}{n^2}\)
(B) \(\dfrac {\lambda}{n}\)
(C) \(\dfrac {\lambda}{\sqrt{n}}\)
(D) \(n \lambda\)
(E) \(\sqrt{n} \lambda\)
(F) \(\lambda\)
QUESTION 02
Find the relationship between \(\theta\) and \(phi\).
(A) \(\tan \theta = n \tan \phi\)
(B) \(\cos \theta = n^2 \cos \phi\)
(C) \(\cos \theta = n \cos \phi\)
(D) \(\sin \theta = n \sin \phi\)
(E) \(\sin \theta = \dfrac{\sin \phi}{n}\)
(F) \(\cos \theta = \dfrac{\cos \phi}{n}\)
QUESTION 03
If a certain condition is satisfied, constructive interference occurs between the light reflected at C and the light traveling through the path ABC. The condition is expressed using a non-negative integer, \(m\). Find this condition.
(A) \(QB + BC = \left (m + \dfrac 12 \right) \dfrac {\lambda}{n}\)
(B) \(AB + PC = m \lambda\)
(C) \(QB + BC = m \lambda\)
(D) \(AB + BC = \left (m + \dfrac 12 \right) \dfrac {\lambda}{n}\)
QUESTION 04
Find the expression of \(d\) when the condition in (3) is satisfied, and constructive interference occurs.
(A) \(\dfrac {m \lambda}{2n \cos \phi}\)
(B) \(\dfrac {(2m + 1) \lambda}{4n \cos \phi}\)
(C) \(\dfrac {(2m + 1) \lambda}{2n \cos \phi}\)
(D) \(\dfrac {(2m + 1) \lambda}{4 \cos \phi}\)
(E) \(\dfrac {m \lambda}{4n \sin \phi}\)
(F) \(\dfrac {m \lambda}{2 \cos \phi}\)