I. PART 1
QUESTION 01
An object of mass mis launched horizontally with a speed \(v\) at a height of h above the ground level as shown in figure below.
Let \(\theta\) be the impact angle to the ground and \(g\) be the acceleration of gravity. Find the formula of \(\tan \theta\).
(A) \(\dfrac {\sqrt{2gh}}{v}\)
(B) \(\dfrac {\sqrt{gh}}{2v}\)
(C) \(\dfrac {v}{\sqrt{2gh}}\)
(D) \(\dfrac {2v}{\sqrt{gh}}\)
(E) \(\dfrac {\sqrt{gh}}{v}\)
(F) \(\dfrac {v}{\sqrt{gh}}\)
QUESTION 02
An object of mass \(m\) is attached to a light spring with a force constant \(k\) and a natural length \(l_o\). The object is moving on a frictionless flat horizontal table with a uniform circular motion as shown in figure below. The center of the circle \(O\) is at the other end of the spring. During this motion, the length of the spring is extended by \(\alpha l_o ( \alpha > 0)\) from the natural length. Find the speed \(v\) of the object.
(A) \(\sqrt{\dfrac{(1 + \alpha) \alpha m}{k}} l_o\)
(B) \(\sqrt{\dfrac{m}{k}} (1 + \alpha) l_o\)
(C) \(\sqrt{\dfrac{m}{k}} \alpha l_o\)
(D) \(\sqrt{\dfrac{k}{m}} (1 + \alpha) l_o\)
(E) \(\sqrt{\dfrac{(1 + \alpha) \alpha k}{m}} l_o\)
(F) \(\sqrt{\dfrac{k}{m}} \alpha l_o\)
QUESTION 03
A charged particle of mass \(m\) and charge \(q\) is in a uniform electric field \(E\). Initially the particle is at rest, and then accelerated by the electric field. Find the time for the particle to travel at a distance of \(d\) from the initial location.
(A) \(\dfrac{md}{2qE}\)
(B) \(\sqrt{\dfrac{md}{qE}}\)
(C) \(\dfrac{md}{qE}\)
(D) \(\sqrt{\dfrac{md}{2qE}}\)
(E) \(\dfrac{md}{qE}\)
(F) \(\sqrt{\dfrac{2md}{qE}}\)
QUESTION 04
A screen is placed at a large distance \(L\) from a plate where two slits \(\text{S}_{\text{1}}\) and \(\text{S}_{\text{2}}\) are notched. These slits are separated by a distance of \(d\) as shown in figure below. A monochromatic light from a single slit \(\text{S}_{\text{0}}\) with a wavelength of \(\alpha\) passes through the two slits \(\text{S}_{\text{1}}\) and \(\text{S}_{\text{2}}\). Bright and dark interference fringes appear on the screen. Find the distance from the screen center \(\text{O}\) to the third dark line.
(A) \(\dfrac{L \lambda}{d}\)
(B) \(\dfrac{2 L \lambda}{d}\)
(C) \(\dfrac{3 L \lambda}{d}\)
(D) \(\dfrac{L \lambda}{2 d}\)
(E) \(\dfrac{3 L \lambda}{2 d}\)
(F) \(\dfrac{5 L \lambda}{2 d}\)
QUESTION 05
An observer is moving away at a constant speed of 5 m/s from a speaker which is emitting sound waves at a frequency of 660 Hz. The sound speed is 330 m/s. When the sound source S and the observation point O are located as shown in figure below, what frequency of the sound will the observer hear?
(A) 650 Hz
(B) 652 Hz
(C) 654 Hz
(D) 660 Hz
(E) 666 Hz
(F) 668 Hz
II. PART 2
The region on the right hand side of line \(XY\) is filled with a uniform magnetic field of flux density \(B\) pointing out from back to front as shown in figure below. A square coil abed with a side length of \(l\) enters the magnetic field region with a constant speed \(v\). Line \(XY\) and side ab are parallel to each other. The resistance of the coil is \(R\). At a time \(t = 0\), side ab of the coil passes line \(XY\). Answer the following questions in the case of \(0 < t < \dfrac lv\).
QUESTION 01
Find the magnitude of the magnetic flux which passes through the coil at the time \(t\).
(A) \(Blt\)
(B) \(Bl^2t\)
(C) \(vBlt\)
(D) \(vB^2lt\)
(E) \(vBl^2t\)
(F) \(vBt\)
QUESTION 02
Find the magnitude of the electromotive force induced in the coil.
(A) \(vB\)
(B) \(vBl\)
(C) \(vBlt\)
(D) \(Bl\)
(E) \(Bl^2\)
(F) \(vBl^2\)
QUESTION 03
Find the induced current which flows in the coil.
(A) \(\dfrac{vBl^2}{R}\)
(B) \(\dfrac{Bl}{R}\)
(C) \(\dfrac{vB^2l}{R}\)
(D) \(\dfrac{vB}{R}\)
(E) \(\dfrac{vBl}{R}\)
(F) \(\dfrac{Bl^2}{R}\)
QUESTION 04
Find the direction of the induced current which flows in the coil.
(A) \(a -> b -> c -> d\)
(B) \(a -> d -> c -> b\)
QUESTION 05
Find the magnitude of the external force to maintain the constant speed \(v\) of the coil.
(A) \(\dfrac {B^2l}{R}\)
(B) \(\dfrac {vBl^2}{R}\)
(C) \(\dfrac {Bl^2}{R}\)
(D) \(\dfrac {vB^2l^2}{R}\)
(E) \(\dfrac {vB^2l}{R}\)
(F) \(\dfrac {B^2l^2}{R}\)
III. PART 3
At the surface of the earth the acceleration of gravity has the value \(g = 9.8 \text{ m}/\text{s}^2\). The constant of universal gravitation is given by \(G = 6.67 \times 10^{-11} \: \text{N m}^2 /\text{kg}^2\).
QUESTION 01
The radius of the earth is \(6.4 x 10^3 \text{ km}\). Find the mass of the earth using the values of \(g\) and \(G\).
(A) \(6 \times 10^{24} \text{ kg}\)
(B) \(6 \times 10^{18} \text{ kg}\)
(C) \(6 \times 10^{30} \text{ kg}\)
(D) \(2 \times 10^{30} \text{ kg}\)
(E) \(2 \times 10^{36} \text{ kg}\)
(F) \(2 \times 10^{24} \text{ kg}\)
QUESTION 02
An object can escape from the gravitational attraction of a planet if the object has a large enough speed. The minimum value of this speed is called the escape speed. Find the escape speed of the earth.
(A) \(1.1 \times 10^{2} \text{ m/s}\)
(B) \(1.1 \times 10^{3} \text{ m/s}\)
(C) \(1.1 \times 10^{4} \text{ m/s}\)
(D) \(7.9 \times 10^{2} \text{ m/s}\)
(E) \(7.9 \times 10^{3} \text{ m/s}\)
(F) \(7.9 \times 10^{4} \text{ m/s}\)
QUESTION 03
The mass of Jupiter is about 320 times larger than the earth and the radius of Jupiter is about 11 times larger than the earth. What is the ratio of the escape speed from Jupiter to that of the earth?
(A) 2.1
(B) 5.4
(C) 11
(D) 18
(E) 29
(F) 320
QUESTION 04
A satellite moves in a circular orbit around the earth. If the satellite's orbital period is equal to the Earth's rotational period, what is the radius of the satellite's orbit?
(A) \(4.2 \times 10^6 \text{ m}\)
(B) \(4.2 \times 10^7 \text{ m}\)
(C) \(4.2 \times 10^8 \text{ m}\)
(D) \(6.4 \times 10^6 \text{ m}\)
(E) \(6.4 \times 10^7 \text{ m}\)
(F) \(6.4 \times 10^8 \text{ m}\)
IV. PART 4
One mole of a monatomic ideal gas is taken through the cycle shown in figure below. In the process AB the gas pressure increases from \(P_o\) to \(4P_o\) at constant volume \(V = V_o\) . In the process BC the gas volume increases from \(V_o\) to \(4V_o\) at constant pressure \(P = 4P_o\) . In the process CD the gas pressure decreases from \(4P_o\) to \(P_o\) at constant volume \(V = 4V_o\). In the process DA the gas volume decreases from \(4V_o\) to \(V_o\) at constant pressure \(P = P_o\). The gas has a molar specific heat at constant volume, \(C_v = \dfrac 32 R\) with \(R\) the universal gas constant.
QUESTION 01
Find the thermal energy transferred into the system in the process AB.
(A) \(P_o V_o\)
(B) \(3 P_o V_o\)
(C) \(\dfrac {11}{2} P_o V_o\)
(D) \(\dfrac 95 P_o V_o\)
(E) \(\dfrac 92 P_o V_o\)
(F) \(\dfrac 72 P_o V_o\)
QUESTION 02
Find the thermal energy transferred into the system in the process BC.
(A) \(18 P_o V_o\)
(B) \(24 P_o V_o\)
(C) \(30 P_o V_o\)
(D) \(\dfrac {25}{2} P_o V_o\)
(E) \(\dfrac {45}{2} P_o V_o\)
(F) \(\dfrac {75}{2} P_o V_o\)
QUESTION 03
Find the net work done by the gas per cycle.
(A) \(16 P_o V_o\)
(B) \(4 P_o V_o\)
(C) \(3 P_o V_o\)
(D) \(12 P_o V_o\)
(E) \(15 P_o V_o\)
(F) \(9 P_o V_o\)
QUESTION 04
Find the thermal efficiency of the cycle.
(A) \(\dfrac {3}{10}\)
(B) \(\dfrac {3}{5}\)
(C) \(\dfrac {2}{5}\)
(D) \(\dfrac {6}{23}\)
(E) \(0\)
(F) \(1\)
V. PART 5
A sound wave travels in the positive x-direction. As the sound wave propagates the air pressure P changes above and below the normal atmospheric pressure, \(P_o\) . At \(x = 0, \Delta P = P - P_o\) varies time-dependently in a sinusoidal manner as shown in figure below. The horizontal axis represents time \(t\).
QUESTION 01
Find the amplitude of \(\Delta P\).
(A) \(\dfrac A3\)
(B) \(\dfrac A2\)
(C) \(A\)
(D) \(2A\)
(E) \(3A\)
(F) \(4A\)
QUESTION 02
Find the period of the oscillation.
(A) \(\dfrac {t_2 - t_1}{2}\)
(B) \(t_2 - t_1\)
(C) \(2(t_2 - t_1)\)
(D) \(3(t_2 - t_1)\)
(E) \(4(t_2 - t_1)\)
(F) \(5(t_2 - t_1)\)
QUESTION 03
The speed of the sound wave is \(v\). Find the wavelength of the sound wave.
(A) \(2v (t_2 - t_1)\)
(B) \(v (t_2 - t_1)\)
(C) \(\dfrac {v (t_2 - t_1)}{2}\)
(D) \(\dfrac {v (t_2 - t_1)}{\pi}\)
(E) \(\dfrac {v (t_2 - t_1)}{2 \pi}\)
(F) \(\dfrac {v (t_2 - t_1)}{4 \pi}\)
QUESTION 04
A snapshot of \(\Delta P\) as a function of \(x\) at a certain time \(t\) is shown in Fig. Which is the highest density point?
(A) a
(B) b
(C) c
(D) d