MikaStars39/nano-eval huggingface dataset:
Assume that the collision between gas molecules and the walls of the container is inelastic. During the collision, the molecules can transfer only a portion $\alpha$ of their energy to the walls. What is the energy transferred by the gas to the unit area of the wall in unit time ? The temperature of the gas is $T$, the molecular mass is $m$, and the number density of the molecules is $n$ .

Options:
A. $$
\alpha\, \frac{\pi k T n} {3} \left( \frac{2 m} {\pi} \right)^{3 / 2}
$$
B. $$
\alpha\, \frac{\pi m n} {2} \left( \frac{2 k T} {\pi m} \right)^{3 / 2}
$$
C. $$
\alpha\, \frac{2 \pi n} {m} \left( \frac{k T} {2 \pi m} \right)^{3 / 2}
$$
D. $$
\alpha\, \frac{\pi m} {2 n} \left( \frac{2 k T} {m \pi} \right)^{3 / 2}
$$
E. $$
\alpha\, \frac{\pi n m} {4} \left( \frac{k T} {\pi m} \right)^{3 / 2}
$$
F. $$
\alpha\, \frac{\pi n} {2 m} \left( \frac{2 k \pi T} {m} \right)^{3 / 2}
$$
G. $$
\alpha\, \frac{2 m n} {\pi} \left( \frac{k T} {m \pi} \right)^{3 / 2}
$$
H. $$
\alpha\, \frac{\pi n T} {m} \left( \frac{2 k m} {\pi} \right)^{3 / 2}
$$
I. $$
\alpha\, \frac{2 \pi k T} {m n} \left( \frac{n m} {\pi} \right)^{3 / 2}
$$
J. $$
\alpha\, \frac{\pi n T} {2 m} \left( \frac{k 2 m} {\pi} \right)^{3 / 2}
$$
Answer: B
A building within a height control area has an outdoor ground elevation of 10.300 meters, a roof surface elevation of 18.000 meters, and a parapet top elevation of 19.100 meters. The highest point of the building is the top surface of the water tank protruding from the roof, with an elevation of 21.300 meters. The building's height is () meters.

Options:
A. 20.100
B. 18.800
C. 19.600
D. 21.600
E. 19.900
F. 20.800
G. 20.500
H. 22.000
I. 19.400
J. 21.100
Answer: D
A certain project requires excavation of $4,800 \mathrm{m}^{3}$ of earth, which is organized into four sections for construction. It is planned to use two excavators for the earthwork, with each excavator having a productivity quota of $50 \mathrm{m}^{3}$ per machine shift. Two teams will work in shifts for this project, so the flow rhythm of the earthwork excavation is () days.

Options:
A. 8
B. 11
C. 13
D. 9
E. 4
F. 7
G. 12
H. 10
I. 5
J. 6
Answer: J
A uniformly charged oil-filled electric field gap has a distance of $30 \mathrm{mm}$, with an alternating voltage of 300kV applied between the electrodes. Assume the permittivity of the oil is $\varepsilon_{\mathrm{r1}}=2$ and that of the barrier is $\varepsilon_{r2}=4$. If a single barrier with a thickness of 3mm or three such barriers are placed between the electrodes, by how many times is the electric field intensity in the oil increased compared to when there are no barriers?

Options:
A. 1.8000
B. 0.9500
C. 1.4000
D. 1.3000
E. 1.1765
F. 1.2500
G. 1.5000
H. 2.0000
I. 1.0500
J. 1.1000
Answer: E
A cable is 100 meters long, with the inner and outer radii of the insulation being 5 cm and 15 cm, respectively. At 20°C, the volume resistivity of the insulation is $\rho_{\mathrm{v}} = 3 \times 10^{12} \, \Omega \, \cdot \, \mathrm{cm}$, and the resistance temperature coefficient is $\alpha = 0.02^{\circ} \mathrm{C}^{-1}$. If the cable length is 200 meters, what is the volume insulation resistance at 20°C?

Options:
A. $$
7. 9 2 \times1 0^{1 2} \Omega
$$
B. $$
6. 4 9 \times1 0^{1 3} \Omega
$$
C. $$
1. 9 6 \times1 0^{1 3} \Omega
$$
D. $$
9. 5 4 \times1 0^{1 3} \Omega
$$
E. $$
5. 2 7 \times1 0^{1 2} \Omega
$$
F. $$
8. 0 5 \times1 0^{1 2} \Omega
$$
G. $$
2. 7 1 \times1 0^{1 4} \Omega
$$
H. $$
1. 8 3 \times1 0^{1 4} \Omega
$$
I. $$
4. 6 5 \times1 0^{1 3} \Omega
$$
J. $$
3. 0 8 \times1 0^{1 2} \Omega
$$
Answer: D
A bimetallic strip is composed of two metal strips with linear expansion coefficients $\alpha_{1}$ and $\alpha_{2}$, both having a thickness of $d$. At temperature $T_{0}$, they both have a length of $L_{0}$. When the temperature changes by $\Delta T$, they can bend together, taking on an arc shape. What is the curvature radius $R$ of this arc?

Options:
A. $$
R={\frac{d} {\left( \, \alpha_{1} \,-\, \alpha_{2} \, \right) \alpha_{1}}}$$
B. $$
R={\frac{d} {\left( \, \alpha_{1} \,-\, \alpha_{2} \, \right) \Delta T}}
$$
C. $$
R={\frac{d} {\left( \, \alpha_{1} \,-\, d \, \right) \Delta T}}$$
D. $$
R={\frac{d} {\left( \, \alpha_{1} \,-\, \alpha_{2} \, \right) L_{0}}}$$
E. $$
R={\frac{2d} {\left( \, \alpha_{1} \,-\, \alpha_{2} \, \right) \Delta T}}$$
F. $$
R={\frac{L_{0}} {\left( \, \alpha_{1} \,-\, \alpha_{2} \, \right) \Delta T}}$$
G. $$
R={\frac{d} {\left( \, \alpha_{2} \,-\, \alpha_{1} \, \right) \Delta T}}$$
H. $$
R={\frac{d} {\left( \, \alpha_{1} \,+\, \alpha_{2} \, \right) \Delta T}}$$
I. $$
R={\frac{L_{0}} {\left( \, \alpha_{2} \,-\, \alpha_{1} \, \right) \Delta T}}$$
J. $$
R={\frac{d \Delta T} {\left( \, \alpha_{1} \,-\, \alpha_{2} \, \right)}}$$
Answer: B
Give nitrogen gas with a mass of $10 \times 10^{-3} \text{ kg}$. When the pressure is 0.101 MPa and the volume is $7700 \text{ cm}^{3}$, what is the average translational kinetic energy of the nitrogen molecule ?

Options:
A. $$
4. 0 5 \times1 0^{-2 1} \mathrm{J}
$$
B. $$
3. 9 8 \times1 0^{-2 0} \mathrm{J}
$$
C. $$
5. 4 2 \times1 0^{-2 1} \mathrm{J}
$$
D. $$
9. 6 7 \times1 0^{-1 9} \mathrm{J}
$$
E. $$
1. 8 9 \times1 0^{-2 3} \mathrm{J}
$$
F. $$
4. 7 2 \times1 0^{-2 2} \mathrm{J}
$$
G. $$
2. 5 4 \times1 0^{-2 3} \mathrm{J}
$$
H. $$
7. 0 9 \times1 0^{-2 0} \mathrm{J}
$$
I. $$
8. 1 4 \times1 0^{-2 2} \mathrm{J}
$$
J. $$
6. 3 5 \times1 0^{-1 9} \mathrm{J}
$$
Answer: C
Given that the ultracentrifuge rotates at an angular velocity of $\omega$, with a colloid density of $\rho$ and solvent density $\rho_{0}$, and the measured ratio of particle concentration at distances $r_{2}$ and $r_{1}$ from the axis of the centrifuge is $\alpha$. What is the molar mass $M$ of the colloid molecules?

Options:
A. $$
\frac{2 \ R \ T \mathrm{l n} \ \eta} {\left( 1-\frac{\rho_{0}} {\rho} \right) \ \omega^{2} ( r_{2}^{2} \,-\, r_{1}^{2} )}
$$
B. $$
\frac{2 \ R \ \mathrm{T} \ \ln \ \alpha} {\left( 1-\frac{\rho_{0}} {\rho} \right) \omega \, ( r_{2}^{2} \,-\, r_{1}^{2} )}
$$
C. $$
\frac{2 \ R \mathrm{T} \ln^2 \ \alpha} {\left( 1-\frac{\rho_{0}} {\rho} \right) \ \omega^{2} ( r_{2}^{2} \,-\, r_{1}^{2} )}
$$
D. $$
\frac{3 \ R \mathrm{T} \ \ln \ \alpha} {\left( 1+\frac{\rho_{0}} {\rho} \right) \ \omega^{2} ( r_{2}^{2} \,-\, r_{1}^{2} )}
$$
E. $$
\frac{2 \ R \ \mathrm{T} \ \ln \ \alpha} {\left( 1-\frac{\rho_{0}} {\rho} \right) \ \omega^{2} \ ( r_{2} \,-\, r_{1} )}
$$
F. $$
\frac{2 \ R\ T \ln \ \alpha} {\left( 1-\frac{\rho} {\rho_{0}} \right) \ \omega^{2} ( r_{2}^{2} \,-\, r_{1}^{2} )}
$$
G. $$
\frac{2 \ R \ \mathrm{T} \ \ln \ \alpha} {\left( 1-\frac{\rho_{0}} {\rho} \right) \ \omega^{2} ( r_{2}^{3} \,-\, r_{1}^{3} )}
$$
H. $$
\frac{R T \mathrm{l n} \ \alpha}{\left( 1-\frac{\rho_{0}} {\rho} \right) \ \omega^{2} ( r_{1}^{2} \,-\, r_{2}^{2} )}
$$
I. $$
\frac{2 R T \mathrm{l n} \ \alpha} {\left( 1-\frac{\rho_{0}} {\rho} \right) \omega^{2} ( r_{2}^{2} \,-\, r_{1}^{2} )}
$$
J. $$
\frac{2 \ R \ T \ln \ \beta} {\left( 1-\frac{\rho_{0}} {\rho} \right) \ \omega^{3} ( r_{2}^{2} \,-\, r_{1}^{2} )}
$$
Answer: I
An electron beam emitted by an electron gun enters a gas with a pressure of $p$. A collecting electrode is placed at a distance $x$ from the electron gun, used to measure the number of electrons that can freely pass through (i.e., not collide with gas molecules) over this distance. It is known that the emission current of the electron gun is $100 \, \mu A$. When the gas pressure $p_{1} = 100 \, \mathrm{Pa}$ and $x = 10 \, \mathrm{cm}$, the current reaching the collecting electrode is $37 \, \mu A$. What is the current reaching the collecting electrode when the pressure drops to $p_{2} = 50 \, \mathrm{Pa}$?

Options:
A. $$
6 1 \ \mu\mathrm{A}
$$
B. $$
7 3 \ \mu\mathrm{A}
$$
C. $$
4 8 \ \mu\mathrm{A}
$$
D. $$
3 5 \ \mu\mathrm{A}
$$
E. $$
5 0 \ \mu\mathrm{A}
$$
F. $$
8 5 \ \mu\mathrm{A}
$$
G. $$
2 9 \ \mu\mathrm{A}
$$
H. $$
7 8 \ \mu\mathrm{A}
$$
I. $$
4 2 \ \mu\mathrm{A}
$$
J. $$
9 0 \ \mu\mathrm{A}
$$
Answer: A
The volume of a sealed adiabatic container is $V_{\mathrm{0}}=6.0\times10^{-3} \mathrm{m}^{3}$, filled with saturated vapor at a temperature of $T_{1}=393 \mathrm{K}$ and a saturation vapor pressure of $p_{1}=1.96 \times10^{5}\mathrm{Pa}$. If water at 10°C is injected into the container, the temperature of the water drops to $T_{2}=373\mathrm{K}$, and the corresponding saturation vapor pressure is $p_{2}=9.81 \times 10^{4} \mathrm{Pa}$. What is the mass of the injected water ?
( The specific heat capacity of water is known to be $4. 186\times10^{3}\mathrm{J} \cdot \mathrm{kg}^{-1}\cdot\mathrm{K}^{-1}$, the heat of vaporization is $2. 2 6 \times1 0^{6} \mathrm{J} \cdot \mathrm{kg}^{-1}$, and the molar specific heat capacity at constant volume for water vapor is $G_{V, m}=3 R$, where $R$ is the molar gas constant. )

Options:
A. $$
2 0. 3 \times1 0^{-3} \mathrm{~ k g}
$$
B. $$
2 5. 4 \times1 0^{-3} \mathrm{~ k g}
$$
C. $$
1 7. 2 \times1 0^{-3} \mathrm{~ k g}
$$
D. $$
2 2. 1 \times1 0^{-3} \mathrm{~ k g}
$$
E. $$
9. 9 \times1 0^{-3} \mathrm{~ k g}
$$
F. $$
1 2. 7 \times1 0^{-3} \mathrm{~ k g}
$$
G. $$
1 5. 5 \times1 0^{-3} \mathrm{~ k g}
$$
H. $$
1 6. 3 \times1 0^{-3} \mathrm{~ k g}
$$
I. $$
1 8. 9 \times1 0^{-3} \mathrm{~ k g}
$$
J. $$
2 1. 8 \times1 0^{-3} \mathrm{~ k g}
$$
Answer: I
A rectangular container made of thermal conductive material is divided by a partition into two equal-volume parts, containing equal masses of dry air and moist air, respectively. In the moist air, the mass of water vapor accounts for $2\%$.
If the partition is removed without allowing any leakage, the proportional relationship between the pressure at equilibrium and the individual pressures of the dry and moist air before the partition was removed is ( ) .
( The molar mass of dry air $\mu_{\text{air}} = 28.8 \times 10^{-3} \ \mathrm{kg/mol}$. )

Options:
A. $$
1. 0 0 5 \colon1 \colon1. 0 1 8
$$
B. $$
1. 0 1 5 \colon1 \colon0. 9 8 5
$$
C. $$
1. 0 1 2 \colon1 \colon1. 0 0 6
$$
D. $$
0. 9 5 0 \colon1 \colon1. 1 0 5
$$
E. $$
1. 0 0 6 \colon1 \colon1. 0 1 2
$$
F. $$
1. 0 0 9 \colon1 \colon1. 0 0 8
$$
G. $$
1. 0 0 4 \colon1 \colon1. 0 2 1
$$
H. $$
1. 0 0 7 \colon1 \colon1. 0 3 0
$$
I. $$
1. 0 0 3 \colon1 \colon1. 0 2 5
$$
J. $$
1. 0 1 0 \colon1 \colon1. 0 2 0
$$
Answer: E
From measurements, it is known that the natural frequency of a spring oscillator is 50 Hz. If the mass of the block is increased by $5 \mathrm{g}$, its natural frequency changes to 45 Hz. What is the spring constant ().

Options:
A. Elastic coefficient is 1780.56 N/m.
B. Elastic coefficient is 2120.00 N/m.
C. Elastic coefficient is 2103.78 N/m.
D. Elastic coefficient is 2000.25 N/m.
E. Elastic coefficient is 2203.50 N/m.
F. Elastic coefficient is 1983.45 N/m.
G. Elastic coefficient is 2350.12 N/m.
H. Elastic coefficient is 2155.42 N/m.
I. Elastic coefficient is 2060.31 N/m.
J. Elastic coefficient is 1900.78 N/m.
Answer: C
The mass of the speaker diaphragm is $1 \ g$, and displacing the driving rod 1 mm from the equilibrium position requires a force of $1 \times 10^{6}$ dyne. The frictional resistance to motion is proportional to the diaphragm's velocity, being 300 dyne when the velocity is 1 cm/s. Assuming the diaphragm moves like a spring oscillator, find its natural frequency and the time it takes for its amplitude to decay to $\frac{1}{e}$ of its initial value. If the driving rod is subjected to a driving force of $10^{5} \cos(2 \pi f t)$ dyne, try to plot the real and imaginary parts of the diaphragm's force impedance as a function of frequency, with the frequency ranging from $f = 0 \ \mathrm{Hz}$ to $f = 1000 \ \mathrm{Hz}$.

Options:
A. $$
f_{0} \;=\; 3 8 9 \mathrm{~ H z}, \; \tau\;=\; 8. 4 \mathrm{~ m s }
$$
B. $$
f_{0} \;=\; 4 2 0 \mathrm{~ H z}, \; \tau\;=\; 8. 0 \mathrm{~ m s }
$$
C. $$
f_{0} \;=\; 6 1 2 \mathrm{~ H z}, \; \tau\;=\; 4. 3 \mathrm{~ m s }
$$
D. $$
f_{0} \;=\; 4 9 8 \mathrm{~ H z}, \; \tau\;=\; 6. 1 \mathrm{~ m s }
$$
E. $$
f_{0} \;=\; 5 7 0 \mathrm{~ H z}, \; \tau\;=\; 5. 9 \mathrm{~ m s }
$$
F. $$
f_{0} \;=\; 3 9 5 \mathrm{~ H z}, \; \tau\;=\; 9. 1 \mathrm{~ m s }
$$
G. $$
f_{0} \;=\; 4 3 7 \mathrm{~ H z}, \; \tau\;=\; 7. 9 \mathrm{~ m s }
$$
H. $$
f_{0} \;=\; 5 5 0 \mathrm{~ H z}, \; \tau\;=\; 5. 1 \mathrm{~ m s }
$$
I. $$
f_{0} \;=\; 5 0 4 \mathrm{~ H z}, \; \tau\;=\; 6. 7 \mathrm{~ m s }
$$
J. $$
f_{0} \;=\; 5 8 8 \mathrm{~ H z}, \; \tau\;=\; 5. 5 \mathrm{~ m s }
$$
Answer: I
A circular membrane with a radius of $10 \; \mathrm{cm}$ and a surface density of $1.0 \ \mathrm{kg} / \mathrm{m}^{2}$ is stretched to achieve a uniform tension of $10,000 \; \mathrm{N/m}$. Calculate the three lowest natural frequencies of the membrane undergoing free symmetric vibration.

Options:
A. $$
3 9 0 \mathrm{~ H z}, 8 8 0 \mathrm{~ H z}, 1 3 9 0 \mathrm{~ H z}
$$
B. $$
3 8 3 \mathrm{~ H z}, 8 7 9 \mathrm{~ H z}, 1 3 7 8 \mathrm{~ H z}
$$
C. $$
3 5 0 \mathrm{~ H z}, 8 6 0 \mathrm{~ H z}, 1 3 5 0 \mathrm{~ H z}
$$
D. $$
3 8 7 \mathrm{~ H z}, 8 6 9 \mathrm{~ H z}, 1 4 7 8 \mathrm{~ H z}
$$
E. $$
3 6 5 \mathrm{~ H z}, 8 9 0 \mathrm{~ H z}, 1 3 6 5 \mathrm{~ H z}
$$
F. $$
4 0 0 \mathrm{~ H z}, 9 0 0 \mathrm{~ H z}, 1 4 0 0 \mathrm{~ H z}
$$
G. $$
3 7 0 \mathrm{~ H z}, 8 8 5 \mathrm{~ H z}, 1 3 7 5 \mathrm{~ H z}
$$
H. $$
3 8 8 \mathrm{~ H z}, 8 7 5 \mathrm{~ H z}, 1 3 6 8 \mathrm{~ H z}
$$
I. $$
3 8 5 \mathrm{~ H z}, 8 8 7 \mathrm{~ H z}, 1 2 7 8 \mathrm{~ H z}
$$
J. $$
3 8 2 \mathrm{~ H z}, 9 7 9 \mathrm{~ H z}, 1 3 8 9 \mathrm{~ H z}
$$
Answer: B
If the frequency of a plane sound wave is 1000 Hz and the displacement amplitude is $1 0^{-5} \text{ m}$, try to calculate the sound intensity under standard atmospheric pressure and temperature (20°C).

Options:
A. $$
0. 5 8 \ \mathbf{W} / \mathbf{m}^{2}
$$
B. $$
0. 8 2 \ \mathbf{W} / \mathbf{m}^{2}
$$
C. $$
0. 9 1 \ \mathbf{W} / \mathbf{m}^{2}
$$
D. $$
0. 9 8 \ \mathbf{W} / \mathbf{m}^{2}
$$
E. $$
1. 1 6 \ \mathbf{W} / \mathbf{m}^{2}
$$
F. $$
1. 0 5 \ \mathbf{W} / \mathbf{m}^{2}
$$
G. $$
1. 3 4 \ \mathbf{W} / \mathbf{m}^{2}
$$
H. $$
0. 7 5 \ \mathbf{W} / \mathbf{m}^{2}
$$
I. $$
1. 4 0 \ \mathbf{W} / \mathbf{m}^{2}
$$
J. $$
0. 6 7 \ \mathbf{W} / \mathbf{m}^{2}
$$
Answer: B
Bats can use the ultrasonic pulses they emit to navigate. Assuming the bat's sound emission frequency is 39000 Hz, during a flight directly toward a flat wall, its speed is $\frac{1}{40}$ of the speed of sound in air. What is the frequency of the pulse wave reflected back from the wall that the bat hears?

Options:
A. $$
38000 \mathrm{H z}.
$$
B. $$
43000 \mathrm{H z}.
$$
C. $$
40000 \mathrm{H z}.
$$
D. $$
39000 \mathrm{H z}.
$$
E. $$
42000 \mathrm{H z}.
$$
F. $$
37000 \mathrm{H z}.
$$
G. $$
41000 \mathrm{H z}.
$$
H. $$
36000 \mathrm{H z}.
$$
I. $$
39500 \mathrm{H z}.
$$
J. $$
44000 \mathrm{H z}.
$$
Answer: G
In an infinitely long smooth-walled tube with a radius of 2 cm, a plane wave with a frequency of 300 Hz and a power of 0.1 $\mathbf{W}$ propagates along the axis of the tube. If there is a small hole with a radius of 0.5 cm on the tube wall, find: the reflected sound power; the transmitted sound power; and the sound power passing through the small hole.

Options:
A. $$ 0. 0 2 8 8 \ {\bf w} ;0. 0 0 8 2 \ {\bf w} ;0 \ {\bf w} $$
B. $$ 0. 0 5 9 8 \ {\bf w} ;0. 0 0 5 2 \ {\bf w} ;0 \ {\bf w} $$
C. $$ 0. 0 7 9 8 \ {\bf w} ;0. 0 0 1 2 \ {\bf w} ;0 \ {\bf w} $$
D. $$ 0. 0 3 9 8 \ {\bf w} ;0. 0 0 7 2 \ {\bf w} ;0 \ {\bf w} $$
E. $$ 0. 0 1 9 8 \ {\bf w} ;0. 0 0 9 2 \ {\bf w} ;0 \ {\bf w} $$
F. $$ 0. 0 6 9 8 \ {\bf w} ;0. 0 0 4 2 \ {\bf w} ;0 \ {\bf w} $$
G. $$ 0. 0 4 7 8 \ {\bf w} ;0. 0 0 6 2 \ {\bf w} ;0 \ {\bf w} $$
H. $$ 0. 0 0 5 8 \ {\bf w} ;0. 0 0 1 0 \ {\bf w} ;0 \ {\bf w} $$
I. $$
0. 0 9 7 8 \ {\bf w} ;0. 0 0 2 2 \ {\bf w} ;0 \ {\bf w}
$$
J. $$ 0. 0 8 7 8 \ {\bf w} ;0. 0 0 3 2 \ {\bf w} ;0 \ {\bf w} $$
Answer: I
The radius of the piston sound source on an infinite baffle is 10 cm, and the equivalent mass of its mechanical system is 1000 g. It is known that its resonance frequency when operating in water is 2000 Hz. What is the equivalent elastic modulus of this system? What change occurs to the resonance frequency of this piston source when it operates in air?

Options:
A. 3.33 × 10 $^{8}$ N/m; the resonant frequency decreased.
B. 9.88 × 10 $^{8}$ N/m; the resonant frequency remained constant.
C. 5.17 × 10 $^{7}$ N/m; the resonant frequency remained constant.
D. 6.12 × 10 $^{8}$ N/m; the resonant frequency decreased.
E. 5.17 × 10 $^{8}$ N/m; the resonant frequency has increased.
F. 1.23 × 10 $^{8}$ N/m; the resonant frequency decreased.
G. 2.99 × 10 $^{9}$ N/m; the resonant frequency decreased.
H. 7.05 × 10 $^{8}$ N/m; the resonant frequency decreased.
I. 4.22 × 10 $^{9}$ N/m; the resonant frequency remained constant.
J. 8.44 × 10 $^{8}$ N/m; the resonant frequency remained constant.
Answer: E
A direct radiation-type loudspeaker is embedded in an infinite baffle, with a radius of 0.2 m and a mass of 0.04 $\mathbf{kg}$. The voice coil has a resistance of 4 $\Omega$ and an inductance of 0.1 mH, with a transducer coefficient of 10 $\mathrm{T} \cdot \mathrm{m}$. The stiffness of the cone suspension system is 2000 N/m, and the mechanical resistance is 2 $\mathrm{N} \cdot \mathrm{s} / \mathrm{m}$. If an AC voltage with a frequency of 200 Hz and an RMS value of 10 V is applied to the voice coil, assuming there is a radiation load only on one side of the cone, calculate the sound power output of the loudspeaker. Assuming the sound radiation pattern of the loudspeaker is the same as that of a circular piston on an infinite baffle, determine the axial sound pressure level at a distance of 10 m from the loudspeaker.

Options:
A. $$ 3.0 \ \mathbf{W} ; 1 0 0 \ \mathrm{d B} $$
B. $$
1 \ \mathbf{W} ;8 8 \ \mathrm{d B}
$$
C. $$ 1.5 \ \mathbf{W} ; 9 1 \ \mathrm{d B} $$
D. $$ 2.0 \ \mathbf{W} ; 9 5 \ \mathrm{d B} $$
E. $$ 1.8 \ \mathbf{W} ; 9 3 \ \mathrm{d B} $$
F. $$ 0.5 \ \mathbf{W} ; 7 5 \ \mathrm{d B} $$
G. $$ 2.5 \ \mathbf{W} ; 9 8 \ \mathrm{d B} $$
H. $$ 0.8 \ \mathbf{W} ; 7 9 \ \mathrm{d B} $$
I. $$ 0.3 \ \mathbf{W} ; 7 0 \ \mathrm{d B} $$
J. $$ 0.2 \ \mathbf{W} ; 6 8 \ \mathrm{d B} $$
Answer: B
In dynamic conditions, the alternating magnetostrictive stress $T=\sigma(B_{\mathrm{0}})B$, reaches its maximum value of $2.0 \times 10^{4}$ dyne/G·cm$^{2}$ for nickel when $B_{\mathrm{0}}=5100$ G. Determine the maximum magnetic induction $B$ that does not exceed the yield strength.

Options:
A. 2.35×10 $^{4}$G
B. 2.50×10 $^{4}$G
C. 2.32×10 $^{4}$G
D. 2.48×10 $^{4}$G
E. 2.42×10 $^{4}$G
F. 2.30×10 $^{4}$G
G. 2.55×10 $^{4}$G
H. 2.60×10 $^{4}$G
I. 2.39×10 $^{4}$G
J. 2.45×10 $^{4}$G
Answer: E
The operating frequency of a single sonar transducer is 30kHz, and the sound source level is 210dB (referenced to 1μPa/m plane wave sound pressure). Assuming spherical spreading, what are the sound pressure levels at 1000m and 5000m? At what distance is the sound pressure level 100dB? At what distance is the loss due to spherical spreading equal to the loss caused by seawater absorption?

Options:
A. 148 dB,105.0 dB,8000 m,15000 m
B. 140 dB,109.0 dB,9000 m,16000 m
C. 145 dB,108.5 dB,7000 m,15000 m
D. 147 dB,104.5 dB,10000 m,13300 m
E. 146 dB,106.0 dB,8000 m,15500 m
F. 139 dB,108.0 dB,6000 m,15200 m
G. 144 dB,107.5 dB,6000 m,14610 m
H. 142 dB,105.5 dB,6000 m,14500 m
I. 141 dB,107.0 dB,5000 m,14700 m
J. 143 dB,106.5 dB,5000 m,14000 m
Answer: G
In a sea where the speed of sound is $c_{\mathrm{0}}$ = 1500 m/s and there is a constant sound speed gradient of $g = -0.5/\mathrm{s}$, a sonar located on the surface is used to detect an underwater target at a depth of 50 m. Assume the target is found when the transducer array's depression angle is 60°. Find the horizontal distance, the straight-line distance between the target and the base array, and the error produced when the curvature of the sound path is ignored during distance measurement.

Options:
A. 28.8m,57.3m,2.3%
B. 28.4m,57.0m,1.7%
C. 27.9m,55.8m,1.2%
D. 27.6m,55.0m,1.8%
E. 29.1m,58.0m,1.5%
F. 28.6m,57.6m,0.3%
G. 28.0m,56.5m,2.0%
H. 30.0m,59.2m,0.8%
I. 29.5m,58.5m,0.9%
J. 30.2m,59.8m,1.1%
Answer: F
A destroyer is equipped with a detection transducer on its keel, pointing vertically towards the seabed, operating at a frequency of 10 kHz, with a source level of 180 dB and a directivity index (DI) of 15 dB. Assuming a seabed reflection loss of 15 dB and a depth sounder receiver bandwidth of 500 Hz, with a sea depth of 1.0x10 $^{4}$ ft. Determine the speed at which the seabed echo level equals the self-noise level.

Options:
A. 25kn
B. 35kn
C. 28kn
D. 22kn
E. 20kn
F. 10kn
G. 24kn
H. 18kn
I. 30kn
J. 15kn
Answer: E
Assume the depth of the sound channel in shallow water at the same sound speed is 50m, and the frequency of the sound wave is 1kHz. Estimate the transmission loss when the propagation distance is 10km. The seabed is liquid, with a density 1.5 times that of seawater. The speed in the seabed is 1700m/s, and the sound speed in seawater is 1500m/s. Both the sound source and receiver are at a depth of 25m. It is assumed that within the above distance range, the contribution of only the lowest mode is significant.

Options:
A. 65.1dB
B. 72.5dB
C. 71.3dB
D. 68.9dB
E. 80.2dB
F. 52.6dB
G. 66.3dB
H. 59.4dB
I. 48.7dB
J. 60.8dB
Answer: G
The dimensions of a reverberation chamber are 2m × 2.5m × 3m, with an average absorption coefficient of 0.036. When the output power of Wanjiyuan is 1.2mW, what is the maximum sound pressure level inside the room? When there are two people in the room (each with an effective absorption area of 0.50m$^{2}$·Sabin), what does the maximum sound pressure level decrease to?

Options:
A. 95.1 dB,92.1 dB
B. 93.8 dB,94.4 dB
C. 99.0 dB,88.5 dB
D. 94.7 dB,93.6 dB
E. 96.3 dB,91.8 dB
F. 92.5 dB,95.3 dB
G. 94.0 dB,93.2 dB
H. 95.4 dB,92.9 dB
I. 98.2 dB,89.7 dB
J. 97.1 dB,90.2 dB
Answer: H
The intensity spectrum of noise within each hertz of bandwidth is given by $I_{\mathrm{1}} = 10^{-6}/f \, \text{W/m}^{2}$, where $f$ is the center frequency of the bandwidth, measured in Hz. Calculate the intensity spectrum levels at 100 Hz, 500 Hz, and 1000 Hz; determine the intensity level over the entire bandwidth between frequencies 100 Hz to 1000 Hz.

Options:
A. 43dB, 36dB, 33dB (relative to 1pW/(m$^{2}$∙Hz)); 67dB (relative to 1pW/m$^{2}$)
B. 41dB, 34dB, 31dB (relative to 1pW/(m$^{2}$∙Hz)); 65dB (relative to 1pW/m$^{2}$)
C. 45dB, 38dB, 35dB (relative to 1pW/(m$^{2}$∙Hz)); 69dB (relative to 1pW/m$^{2}$)
D. 40dB, 33dB, 30dB (relative to 1pW/(m$^{2}$∙Hz)); 64dB (relative to 1pW/m$^{2}$)
E. 36dB, 29dB, 26dB (relative to 1pW/(m$^{2}$∙Hz)); 60dB (relative to 1pW/m$^{2}$)
F. 39dB, 32dB, 29dB (relative to 1pW/(m$^{2}$∙Hz)); 63dB (relative to 1pW/m$^{2}$)
G. 37dB, 30dB, 27dB (relative to 1pW/(m$^{2}$∙Hz)); 61dB (relative to 1pW/m$^{2}$)
H. 42dB, 35dB, 32dB (relative to 1pW/(m$^{2}$∙Hz)); 66dB (relative to 1pW/m$^{2}$)
I. 44dB, 37dB, 34dB (relative to 1pW/(m$^{2}$∙Hz)); 68dB (relative to 1pW/m$^{2}$)
J. 38dB, 31dB, 28dB (relative to 1pW/(m$^{2}$∙Hz)); 62dB (relative to 1pW/m$^{2}$)
Answer: D
Cylindrical float, with 1 cm depth of mercury at the bottom of the cylindrical tube's lower end. The mass of the tube is much smaller than that of the mercury and can be ignored. Find: How long is the submerged part of the float? Why does the float perform vertical oscillations when lightly pressed? What is its natural frequency? What is the value of the frequency?

Options:
A. 12.9cm;≈1.30Hz
B. 12.8cm;≈1.34Hz
C. 14.2cm;≈1.42Hz
D. 13.6cm;≈1.35Hz
E. 13.5cm;≈1.37Hz
F. 13.4cm;≈1.39Hz
G. 13.7cm;≈1.32Hz
H. 13.8cm;≈1.28Hz
I. 14.1cm;≈1.31Hz
J. 14.0cm;≈1.36Hz
Answer: D
To prepare Buffer 31, a 0.2 M pH 5.00 acetate buffer solution, how much solid sodium acetate (trihydrate, molecular weight 136) and 1 M acetic acid solution are required?

Options:
A. 60.5g,180ml
B. 49.1g,240ml
C. 47.6g,250ml
D. 58.9g,230ml
E. 53.2g,215ml
F. 62.7g,190ml
G. 45.3g,300ml
H. 50.2g,210ml
I. 51.4g,222ml
J. 55.8g,205ml
Answer: I
If given
① $\mathbf{C}_{6} \mathbf{H}_{1 2} \mathbf{O}_{6}$ → $2CH_{3}CHOHCOOH$, $\Delta G^{\circ} = -52000 \mathrm{cal}$
② $\mathbf{C}_{6} \mathbf{H}_{1 2} \mathbf{O}_{6} + 6 \mathbf{O}_{2} \longrightarrow 6 \mathbf{C} \mathbf{O}_{2} + 6 \mathbf{H}_{2} \mathbf{O}$, $\Delta G^{\circ} = -68600 \mathrm{cal}$
Find the $\Delta G^{\circ}$ for the complete oxidation of lactic acid to $CO_{2}$ and $H_{2}O$, and when the energy conversion efficiency to ATP is 40%, how many $\mathrm{ATP}$ can be produced in this process?

Options:
A. $$
- 2 5 0 0 0 0 \mathrm{c a l} \cdot\mathrm{m o l}^{-1},12.5
$$
B. $$
- 3 5 0 0 0 0 \mathrm{c a l} \cdot\mathrm{m o l}^{-1},20.0
$$
C. $$
- 3 1 0 0 0 0 \mathrm{c a l} \cdot\mathrm{m o l}^{-1},16.9
$$
D. $$
- 3 2 0 0 0 0 \mathrm{c a l} \cdot\mathrm{m o l}^{-1},18.1
$$
E. $$
- 3 6 0 0 0 0 \mathrm{c a l} \cdot\mathrm{m o l}^{-1},20.7
$$
F. $$
- 3 4 0 0 0 0 \mathrm{c a l} \cdot\mathrm{m o l}^{-1},19.2
$$
G. $$
- 2 8 0 0 0 0 \mathrm{c a l} \cdot\mathrm{m o l}^{-1},15.7
$$
H. $$
- 2 9 0 0 0 0 \mathrm{c a l} \cdot\mathrm{m o l}^{-1},16.3
$$
I. $$
- 3 8 0 0 0 0 \mathrm{c a l} \cdot\mathrm{m o l}^{-1},21.7
$$
J. $$
- 3 1 7 0 0 0 \mathrm{c a l} \cdot\mathrm{m o l}^{-1},17.4
$$
Answer: J
The unlabeled substrate exhibits competitive inhibition relative to the labeled substrate. This displacement phenomenon can be used in a solution known not to contain a true inhibitor to determine the unknown concentration of the substrate. For example, assume that 0.1 ml of a fraction containing unlabeled S is included in a 1.0 ml standard reaction volume, which contains $[ \mathbf{S^{*}} ]=2.3 \times 10^{-5} \mathrm{mol}$ and enough $[ E ]_{t}$ to produce $V_{\mathrm{max}} = 290 \, \mathrm{nmol/min}$. The observed initial velocity is 48,000 counts per minute. What is the concentration of unlabeled S in the fraction?

Options:
A. $$ 1. 5 8 \times1 0^{-4} \mathrm{m o l} / \mathbf{L} $$
B. $$ 2. 4 9 \times1 0^{-5} \mathrm{m o l} / \mathbf{L} $$
C. $$ 2. 5 3 \times1 0^{-4} \mathrm{m o l} / \mathbf{L} $$
D. $$ 2. 8 4 \times1 0^{-6} \mathrm{m o l} / \mathbf{L} $$
E. $$ 3. 2 1 \times1 0^{-4} \mathrm{m o l} / \mathbf{L} $$
F. $$ 2. 9 0 \times1 0^{-4} \mathrm{m o l} / \mathbf{L} $$
G. $$
1. 3 7 \times1 0^{-4} \mathrm{m o l} / \mathbf{L}
$$
H. $$ 1. 0 5 \times1 0^{-4} \mathrm{m o l} / \mathbf{L} $$
I. $$ 1. 9 6 \times1 0^{-3} \mathrm{m o l} / \mathbf{L} $$
J. $$ 4. 7 8 \times1 0^{-4} \mathrm{m o l} / \mathbf{L} $$
Answer: G
The substrate of a certain enzyme is a weakly acidic $A^{-}$ ion $( \mathrm{p} K_{a} = 4.5 )$. The active site of this enzyme contains a protonated histidine residue necessary for its activity $( \mathbf{p} K_{a} = 6.5 )$. What is the optimal pH for this reaction?

Options:
A. 7.0
B. 9.0
C. 5.5
D. 3.5
E. 8.5
F. 8.0
G. 7.5
H. 6.5
I. 6.0
J. 4.0
Answer: C
Weigh $25mg$ of protease powder and prepare it into a $25ml$ enzyme solution. Take $0.1ml$ of this enzyme solution, using casein as the substrate, and determine enzyme activity with the Folin-phenol colorimetric method, which reveals the production of $1500 \mu\mathbf{g}$ tyrosine per hour. Separately, take $2ml$ of the enzyme solution, and measure the protein nitrogen using the Kjeldahl method to be $0.2mg$. If we consider the amount of enzyme producing $1\mu g$ of tyrosine per minute as one active unit, calculate the total protein content and total activity of $1 g$ of enzyme preparation based on the above data.

Options:
A. $$ 0.700 \ \mathbf{g}, 2.3 \times 10^{5} units $$
B. $$ 0.750 \ \mathbf{g}, 1.5 \times 10^{5} units $$
C. $$ 0.530 \ \mathbf{g}, 3.0 \times 10^{5} units $$
D. $$ 0.625 \ \mathbf{g}, 2.0 \times 10^{5} units $$
E. $$ 0.590 \ \mathbf{g}, 2.6 \times 10^{5} units $$
F. $$ 0.520 \ \mathbf{g}, 3.2 \times 10^{5} units $$
G. $$ 0.680 \ \mathbf{g}, 2.1 \times 10^{5} units $$
H. $$ 0.500 \ \mathbf{g}, 2.8 \times 10^{5} units $$
I. $$ 0.600 \ \mathbf{g}, 2.7 \times 10^{5} units $$
J. $$
0.625 \ \mathbf{g}, 2.5 \times 10^{5} units
$$
Answer: J
Add a solution containing 0.5 mg of $\mathbf{D}$-mannose-$^{14}$C (uniformly labeled) with a specific radioactivity of $3.3 \times 10^{6}$ counts per minute $/ \mu$mol to 50 ml of a solution containing an unknown amount of unlabeled mannose. After mixing, isolate the D-mannose present in the form of osazone. The specific radioactivity of the osazone is 14280 counts per minute $/ \mu$mol. Calculate the concentration of unlabeled D-mannose in the original solution.

Options:
A. $$
0. 0 1 5 0 M
$$
B. $$
0. 0 1 8 9 M
$$
C. $$
0. 0 1 2 8 M
$$
D. $$
0. 0 1 3 2 M
$$
E. $$
0. 0 1 4 7 M
$$
F. $$
0. 0 1 6 4 M
$$
G. $$
0. 0 1 4 5 M
$$
H. $$
0. 0 1 9 3 M
$$
I. $$
0. 0 1 7 6 M
$$
J. $$
0. 0 1 1 9 M
$$
Answer: C
The molar extinction coefficients of compound A at 260 nm and 280 nm are 5248 and 3150, respectively. Now, reagent B extracts A. The molar extinction coefficients of $\mathbf{B}$ reagent at 260 nm and 280 nm are 311 and 350, respectively. The absorbance of the extract is $A_{260 \mathrm{nm}}=2.50, A_{280 \mathrm{nm}}=2.00$. Find the concentration of compound A.

Options:
A. 0.00029M
B. 0.00023M
C. 0.00021M
D. 0.00015M
E. 0.00011M
F. 0.00013M
G. 0.00019M
H. 0.00025M
I. 0.00017M
J. 0.00031M
Answer: G
A meter stick (length taken as 1.0m) is stationary relative to the $\mathrm{K}^{\prime}$ system, forming an angle of 30° with the $x^{\prime}$ axis. If the $\mathrm{K}^{\prime}$ system is moving relative to the $\mathrm{K}$ system along the $x$ axis (parallel to the $x^{\prime}$ axis) direction, and the angle between the meter stick and the $x$ axis measured in the $\mathrm{K}$ system is $45^{\circ}$. What is the length of the meter stick as measured in the $\mathrm{K}$ system ?

Options:
A. $$
1. 0 2 ( \mathrm{m} )
$$
B. $$
0. 9 4 ( \mathrm{m} )
$$
C. $$
0. 7 1 ( \mathrm{m} )
$$
D. $$
0. 3 9 ( \mathrm{m} )
$$
E. $$
1. 1 5 ( \mathrm{m} )
$$
F. $$
0. 6 5 ( \mathrm{m} )
$$
G. $$
0. 8 2 ( \mathrm{m} )
$$
H. $$
0. 4 7 ( \mathrm{m} )
$$
I. $$
1. 2 0 ( \mathrm{m} )
$$
J. $$
0. 5 3 ( \mathrm{m} )
$$
Answer: C
A mirror is moving in vacuum at a speed $v$ in the $+x$ direction. A beam of light with a frequency $\nu$ is perpendicularly incident on the mirror's surface from $x=+\infty$. What is the frequency of the reflected light ?

Options:
A. $$
\frac{v-c} {v+c} \nu
$$
B. $$
{\frac{c+v} {c-v}} \nu
$$
C. $$
{\frac{c-v} {c+v}} \nu
$$
D. $$
\frac{c} {c-v} \nu
$$
E. $$
\frac{c+v} {c} \nu
$$
F. $$
{\frac{c-v} {c}} \nu
$$
G. $$
\frac{v+c} {c} \nu
$$
H. $$
{\frac{c} {v}} \nu
$$
I. $$
{\frac{c-v} {v-c}} \nu
$$
J. $$
\frac{c-v} {c} \nu
$$
Answer: B
What are the maximum radius and minimum density of a celestial body with a mass equal to the Sun's mass $(M=2.0 \times 10^{33} \mathrm{g})$ such that light cannot escape from its surface?

Options:
A. $$
\ \mathrm{1. 4 8 \times1 0^{2} m},
$$
$$
1. 4 5 \times1 0^{2 0} \mathrm{k g} / \mathrm{m}^{3}.
$$
B. $$
\ \mathrm{1. 5 2 \times1 0^{3} m},
$$
$$
1. 4 1 \times1 0^{2 0} \mathrm{k g} / \mathrm{m}^{3}.
$$
C. $$
\ \mathrm{1. 3 8 \times1 0^{3} m},
$$
$$
1. 4 7 \times1 0^{2 1} \mathrm{k g} / \mathrm{m}^{3}.
$$
D. $$
\ \mathrm{1. 4 6 \times1 0^{3} m},
$$
$$
1. 5 0 \times1 0^{2 0} \mathrm{k g} / \mathrm{m}^{3}.
$$
E. $$
\ \mathrm{1. 4 8 \times1 0^{4} m},
$$
$$
1. 4 2 \times1 0^{2 0} \mathrm{k g} / \mathrm{m}^{3}.
$$
F. $$
\ \mathrm{1. 4 7 \times1 0^{3} m},
$$
$$
1. 4 8 \times1 0^{2 0} \mathrm{k g} / \mathrm{m}^{3}.
$$
G. $$
\ \mathrm{1. 4 9 \times1 0^{3} m},
$$
$$
1. 4 3 \times1 0^{2 0} \mathrm{k g} / \mathrm{m}^{3}.
$$
H. $$
\ \mathrm{1. 4 8 \times1 0^{3} m},
$$
$$
1. 4 7 \times1 0^{2 0} \mathrm{k g} / \mathrm{m}^{3}.
$$
I. $$
\ \mathrm{1. 5 1 \times1 0^{3} m},
$$
$$
1. 4 0 \times1 0^{2 0} \mathrm{k g} / \mathrm{m}^{3}.
$$
J. $$
\ \mathrm{1. 5 0 \times1 0^{3} m},
$$
$$
1. 3 7 \times1 0^{2 0} \mathrm{k g} / \mathrm{m}^{3}.
$$
Answer: H
Two particles, each with a rest mass of $m_{0}$, are involved in an interaction. One particle is at rest, while the other has a kinetic energy that is six times its rest energy. The two particles undergo a completely inelastic collision. What are the rest mass and velocity of the composite particle ?

Options:
A. $$
4 m_{0}
$$
$$
{\sqrt{2}} c
$$
B. $$
6 m_{0}
$$
$$
{\frac{\sqrt{3}} {3}} c
$$
C. $$
2 m_{0}
$$
$$
{\frac{1} {3}} c
$$
D. $$
4 m_{0}
$$
$$
{\frac{\sqrt{5}} {4}} c
$$
E. $$
3 m_{0}
$$
$$
{\frac{1} {2}} c
$$
F. $$
5 m_{0}
$$
$$
{\frac{1} {2}} c
$$
G. $$
3 m_{0}
$$
$$
{\frac{\sqrt{5}} {2}} c
$$
H. $$
4 m_{0}
$$
$$
{\frac{\sqrt{3}} {2}} c
$$
I. $$
5 m_{0}
$$
$$
{\frac{\sqrt{2}} {2}} c
$$
J. $$
6 m_{0}
$$
$$
{\frac{1} {4}} c
$$
Answer: H
In the inertial frame $\mathrm{K}$, consider the collision of $n$ objects. These objects are sufficiently small and slow, so that the relativistic momentum can be approximated by Newtonian momentum, and their velocities can be linearly superimposed. They can collide simultaneously at a certain point. Assume that the velocities and rest masses before the collision are $V_{\alpha}, \, m_{\alpha} ( \alpha=1, 2, \cdots, n )$, and after the collision, they are $V_{\alpha}^{\prime}, \, m_{\alpha}^{\prime} ( \alpha=1, 2, \cdots, n )$. (All $m_{\alpha}$ and $m_{\alpha}^{\prime}$ are positive numbers.) For given $V$ and $m$, if after the collision the total kinetic energy of all objects reaches a minimum under the conditions of conservation of energy and momentum, it is referred to as the maximally inelastic collision in the $\mathrm{K}$ frame. At this point, the kinetic energy of the objects is
$$
m_{\alpha} c^{2} \left[ \left( 1-\frac{V_{\alpha}^{2}} {c^{2}} \right)^{-1 / 2}-1 \right]
$$
Give the $m_{\alpha}^{\prime}$, $V_{\alpha}^{\prime}$ for the maximally inelastic collision (within the sufficiently small and slow range where they can be determined).

Options:
A. $$\sum{\bar{m}^{\prime}}_{\alpha} / {\sqrt{1-V_{\mathrm{C M}}^{3} / c^{3}}}
$$
$$
\sum\overline{m}^{\prime}_{\alpha} V_{\alpha}
B. $$\sum{m_{\alpha}^{\prime}} \cdot {\sqrt{1-V_{\mathrm{C M}} / c^{2}}}
$$
$$
\sum{\overline{m}}_{\alpha}^{\prime} V_{\alpha}
$$
C. $$\sum{m_{\alpha}} \cdot \sqrt{1-V_{\mathrm{C M}}^{4} / c^{4}}
$$
$$
\sum{\overline{m}}_{\alpha} + V_{\alpha}
$$
D. $$\sum{m_{\alpha}^{\prime}} \cdot \sqrt{1+V_{\mathrm{C M}}^{2} / c^{2}}
$$
$$
\sum{\overline{m}}_{\alpha}^{\prime} V_{\alpha}^2
$$
E. $$\sum{m_{\alpha}} / {\sqrt{1+V_{\mathrm{C M}}^{2} / c^{2}}}
$$
$$
\sum\overline{m}_{\alpha} V_{\alpha}^2
$$
F. $$\sum{m_{\alpha}} + {\sqrt{1-V_{\mathrm{C M}}^{2} / c^{2}}}
$$
$$
\sum{\overline{m}}_{\alpha} V_{\alpha}^3
$$
G. $$\sum{\bar{m}}_{\alpha}^{2} / {\sqrt{1-V_{\mathrm{C M}}^{2} / c^{3}}}
$$
$$
\sum\overline{m}_{\alpha} \cdot V_{\alpha}
$$
H. $$\sum{\bar{m}}_{\alpha} \cdot {\sqrt{1+V_{\mathrm{C M}}^{2} / c^{2}}}
$$
$$
\sum\overline{\bar{m}}_{\alpha} \cdot V_{\alpha}
$$
I. $$\sum\overline{m}_{\alpha} / {1-V_{\mathrm{C M}} / c}
$$
$$
\sum\overline{m} \cdot V_{\alpha}
$$
J. $$\sum{\bar{m}}_{\alpha} / {\sqrt{1-V_{\mathrm{C M}}^{2} / c^{2}}}
$$
$$
\sum\overline{{{{m}}}}_{\alpha} V_{\alpha}
$$
Answer: J
A $\pi$ meson with momentum $5 m_{\pi} c$ undergoes an elastic collision with a proton that is initially at rest $( m_{p}=7 m_{\pi} )$. What is the velocity of the center of momentum frame ?

Options:
A. 0.52c
B. 0.46c
C. 0.28c
D. 0.41c
E. 0.39c
F. 0.35c
G. 0.48c
H. 0.50c
I. 0.37c
J. 0.44c
Answer: D
A particle with mass $m$ has kinetic energy twice its rest energy. It collides with another particle of the same mass to form a new particle. What is the mass of the new particle ?

Options:
A. $$
18 m^{2}
$$
B. $$
4 m^{2}
$$
C. $$
12 m^{2}
$$
D. $$
14 m^{2}
$$
E. $$
20 m^{2}
$$
F. $$
2 m^{2}
$$
G. $$
6 m^{2}
$$
H. $$
16 m^{2}
$$
I. $$
8 m^{2}
$$
J. $$
10 m^{2}
$$
Answer: I
In the universe, there exists blackbody microwave radiation, with the average photon energy $E \approx 10^{-3} \mathrm{eV}$ and a photon density of $\rho = 300/\mathrm{cm}^{3}$. High-energy $\gamma$ photons can collide with these photons to produce electron-positron pairs, with a production cross section of $\sigma = \frac{\sigma_{\mathrm{T}}}{3}$, where $\sigma_{\mathrm{T}} = \frac{8\pi}{3} r_{\mathrm{e}}^{2}$ is the Thomson scattering cross section for non-relativistic electrons, and $r_{e}$ is the classical electron radius. At what energy levels in the universe would high-energy $\gamma$ photons be limited in their lifetime due to this process?

Options:
A. $$
> 2. 6 1 \times1 0^{1 4} ( \mathrm{e V} )
$$
B. $$
4. 1 0 \times1 0^{1 2} ( \mathrm{e V} )
$$
C. $$
5. 2 5 \times1 0^{1 3} ( \mathrm{e V} )
$$
D. $$
1. 3 7 \times1 0^{1 5} ( \mathrm{e V} )
$$
E. $$
9. 3 9 \times1 0^{1 2} ( \mathrm{e V} )
$$
F. $$
7. 1 9 \times1 0^{1 6} ( \mathrm{e V} )
$$
G. $$
1. 1 8 \times1 0^{1 4} ( \mathrm{e V} )
$$
H. $$
8. 4 5 \times1 0^{1 5} ( \mathrm{e V} )
$$
I. $$
3. 8 1 \times1 0^{1 5} ( \mathrm{e V} )
$$
J. $$
6. 0 4 \times1 0^{1 4} ( \mathrm{e V} )
$$
Answer: A
The $\mathrm{K}_{\mathrm{L}}^{0}$ meson $(M = 498 \mathrm{MeV})$ decays in flight into $\pi^{+} \pi^{-} (mc^{2} = 140 \mathrm{MeV})$. The ratio of the momentum of $\mathrm{K}_{\mathrm{L}}^{0}$ to $M$ is $p/M = 1$. What are the maximum transverse momentum and maximum longitudinal momentum of the $\pi$ mesons in the laboratory frame ?

Options:
A. $$
1 8 0 \mathrm{M e V} / c.
$$
$$
4 9 0. 6 ( \mathrm{M e V} )
$$
B. $$
2 3 0 \mathrm{M e V} / c.
$$
$$
5 6 5. 6 ( \mathrm{M e V} )
$$
C. $$
2 1 0 \mathrm{M e V} / c.
$$
$$
5 5 0. 0 ( \mathrm{M e V} )
$$
D. $$
1 8 5 \mathrm{M e V} / c.
$$
$$
5 2 0. 7 ( \mathrm{M e V} )
$$
E. $$
1 9 5 \mathrm{M e V} / c.
$$
$$
5 3 0. 2 ( \mathrm{M e V} )
$$
F. $$
1 6 5 \mathrm{M e V} / c.
$$
$$
4 8 0. 9 ( \mathrm{M e V} )
$$
G. $$
2 2 0 \mathrm{M e V} / c.
$$
$$
5 6 0. 1 ( \mathrm{M e V} )
$$
H. $$
2 0 6 \mathrm{M e V} / c.
$$
$$
5 4 0. 4 ( \mathrm{M e V} )
$$
I. $$
1 7 5 \mathrm{M e V} / c.
$$
$$
5 1 0. 8 ( \mathrm{M e V} )
$$
J. $$
2 4 5 \mathrm{M e V} / c.
$$
$$
5 7 0. 3 ( \mathrm{M e V} )
$$
Answer: H
A particle with mass $M=3.0 \mathrm{GeV}/c^{2}$ and momentum $p=4.0 \mathrm{GeV}/c$ decays along the $x$-axis into two photons. In the rest frame of the particle, the angular distribution of the photons is isotropic, with the maximum and minimum values in the $x$ direction given by $\frac{\mathrm{d} W}{\mathrm{d} \cos \theta^{*}}=\frac{1}{2}$. What are the maximum and minimum values of the momentum of the photons in the $x$ direction in the laboratory frame ?

Options:
A. $$
3. 5 ( \mathrm{G e V} / c )
$$
$$
-0. 5 ( \mathrm{G e V} / c )
$$
B. $$
4. 0 ( \mathrm{G e V} / c )
$$
$$
-1. 5 ( \mathrm{G e V} / c )
$$
C. $$
6. 0 ( \mathrm{G e V} / c )
$$
$$
-0. 5 ( \mathrm{G e V} / c )
$$
D. $$
3. 0 ( \mathrm{G e V} / c )
$$
$$
0. 5 ( \mathrm{G e V} / c )
$$
E. $$
2. 5 ( \mathrm{G e V} / c )
$$
$$
1. 5 ( \mathrm{G e V} / c )
$$
F. $$
5. 5 ( \mathrm{G e V} / c )
$$
$$
0. 0 ( \mathrm{G e V} / c )
$$
G. $$
5. 0 ( \mathrm{G e V} / c )
$$
$$
-1. 0 ( \mathrm{G e V} / c )
$$
H. $$
4. 5 ( \mathrm{G e V} / c )
$$
$$
-0. 5 ( \mathrm{G e V} / c )
$$
I. $$
5. 0 ( \mathrm{G e V} / c )
$$
$$
0. 5 ( \mathrm{G e V} / c )
$$
J. $$
3. 5 ( \mathrm{G e V} / c )
$$
$$
0. 0 ( \mathrm{G e V} / c )
$$
Answer: H
Assume a non-relativistic charged particle moving in a cyclotron experiences significant radiation energy loss. This fact is explained for a particle with known mass, charge, and initial kinetic energy, given that the particle initially performs circular motion in the uniform magnetic field of the accelerator. Express the kinetic energy of the particle as a function of time.

Options:
A. $$
T_{0} \mathrm{e}^{-{\frac{q^{5} B^{2}} {3 \pi t_{0} m^{2} c^{3}}} t}
$$
B. $$
T_{0} \mathrm{e}^{-{\frac{q^{4} B^{2}} {3 \pi t_{1} m^{3} c^{3}}} t}
$$
C. $$
T_{0} \mathrm{e}^{-{\frac{q^{2} E^{2}} {3 \pi t_{0} m^{3} c}} t}
$$
D. $$
T_{0} \mathrm{e}^{-{\frac{q^{4} B^{2}} {3 \pi t_{0} m^{4} c^{3}}} t}
$$
E. $$
T_{0} \mathrm{e}^{-{\frac{q^{4} B^{3}} {3 \pi t_{0} m^{3} c^{3}}} t}
$$
F. $$
T = T_{0} \mathrm{e}^{-{\frac{q^{4} B^{2}} {3 \pi \epsilon_{0} m^{3} c^{3}}} t}
$$
G. $$
T_{0} \mathrm{e}^{-{\frac{q^{3} B} {3 \pi t_{0} m^{3} c^{3}}} t}
$$
H. $$
T_{0} \mathrm{e}^{-{\frac{q B^{2}} {3 \pi t_{0} m^{3} c^{2}}} t}
$$
I. $$
T_{0} \mathrm{e}^{-{\frac{q^{4} B^{2}} {3 m^{3} t_{0} c^{3}}} t}
$$
J. $$
T_{0} \mathrm{e}^{-{\frac{q^{4} C^{2}} {3 \pi t_{0} m^{3} c^{2}}} t}
$$
Answer: F
A uniformly dense spherical symmetrical cavity, with an inner radius $R$, thickness $H$, total mass $M$, and a rotation period $P$-what is the ratio of the aging rates of rock or air at the center, the inner radius, and the surface due to rotational and self-gravitational effects ? (accurate to a first-order approximation)

Options:
A. $$
1 : \left( 1+\frac{2 \pi^{2} H^{3}} {P^{2} c^{2}} \right) : \left[ 1-\frac{G M} {8 H c^{2}}+\frac{3 \pi^{2} R^{2}} {P^{2} c^{2}} \right]
$$
B. $$
1 : \left( 1-\frac{ \pi^{2} H^{2}} {P^{2} c^{2}} \right) : \left[ 1+\frac{G M} {6 H c^{2}}-\frac{3 \pi^{2} R^{2}} {P^{2} c^{2}} \right]
$$
C. $$
1 : \left( 1-\frac{4 \pi^{2} R^{2}} {5P^{2} c^{2}} \right) : \left[ 1+\frac{G M} {5 R c^{2}}-\frac{ \pi^{2} H^{2}} {P^{2} c^{2}} \right]
$$
D. $$
1 : \left( 1+\frac{\pi^{2} R^{3}} {2P^{2} c^{2}} \right) : \left[ 1-\frac{G M} {5 R c^{2}}+\frac{\pi^{2} R^{2}} {P^{2} c^{2}} \right]
$$
E. $$
1 : \left( 1+\frac{3 \pi^{2} H^{2}} {2P^{2} c^{2}} \right) : \left[ 1-\frac{G M} {7 R c^{2}}+\frac{2 \pi^{2} R^{2}} {P^{2} c^{2}} \right]
$$
F. $$
1 : \left( 1+\frac{5 \pi^{2} R^{2}} {3P^{2} c^{2}} \right) : \left[ 1-\frac{G M} {6 R c^{2}}-\frac{ \pi^{2} R^{2}} {2P^{2} c^{2}} \right]
$$
G. $$
1 : \left( 1+\frac{4 \pi^{2} H^{2}} {P^{2} c^{2}} \right) : \left[ 1-\frac{G M} {8 R c^{2}}-\frac{ \pi^{2} R^{2}} {P^{2} c^{2}} \right]
$$
H. $$
1 : \left( 1-\frac{2 \pi^{2} R^{2}} {P^{2} c^{2}} \right) : \left[ 1+\frac{G M} {6 R c^{2}}-\frac{2 \pi^{2} R^{2}} {P^{2} c^{2}} \right]
$$
I. $$
1 : \left( 1-\frac{3 \pi^{2} R^{2}} {2P^{2} c^{2}} \right) : \left[ 1+\frac{G M} {4 R c^{2}}-\frac{ \pi^{2} H^{2}} {P^{2} c^{2}} \right]
$$
J. $$
1 : \left( 1-\frac{ \pi^{2} H^{2}} {3P^{2} c^{2}} \right) : \left[ 1+\frac{G M} {7 H c^{2}}+\frac{ \pi^{2} R^{2}} {P^{2} c^{2}} \right]
$$
Answer: H
When $\mathrm{a}=1956$ and $\mathrm{a}=5040$, find $\varphi\left( \, \mathbf{a} \, \right)$ and $\mathbf{\sigma} ( \mathbf{a} )$.

Options:
A. $$
\begin{array} {l} {{{\varphi\ ( 1 9 5 6 )=6 4 5}}} \\ {{{\sigma( 1 9 5 6 )=6 4 2 4 2 4}}} \\ {{{\varphi\ ( 5 0 4 0 )=1 2 0 0}}} \\ {{{\sigma( 5 0 4 0 )=2 9 0 9 9 83}}} \\ \end{array}
$$
B. $$
\begin{array} {l} {{{\varphi\ ( 1 9 5 6 )=6 4 2}}} \\ {{{\sigma( 1 9 5 6 )=6 3 3 8 1 9}}} \\ {{{\varphi\ ( 5 0 4 0 )=1 1 3 9}}} \\ {{{\sigma( 5 0 4 0 )=2 9 1 5 703}}} \\ \end{array}
$$
C. $$
\begin{array} {l} {{{\varphi\ ( 1 9 5 6 )=6 5 2}}} \\ {{{\sigma( 1 9 5 6 )=6 3 4 917}}} \\ {{{\varphi\ ( 5 0 4 0 )=1 1 6 0}}} \\ {{{\sigma( 5 0 4 0 )=2 9 1 0287}}} \\ \end{array}
$$
D. $$
\begin{array} {l} {{{\varphi\ ( 1 9 5 6 )=6 6 3}}} \\ {{{\sigma( 1 9 5 6 )=6 3 4 2 3 2}}} \\ {{{\varphi\ ( 5 0 4 0 )=1 1 7 2}}} \\ {{{\sigma( 5 0 4 0 )=2 9 2 0 719}}} \\ \end{array}
$$
E. $$
\begin{array} {l} {{{\varphi\ ( 1 9 5 6 )=6 4 8}}} \\ {{{\sigma( 1 9 5 6 )=6 3 3 7 4 4}}} \\ {{{\varphi\ ( 5 0 4 0 )=1 1 5 2}}} \\ {{{\sigma( 5 0 4 0 )=2 9 0 3 0 4 0}}} \\ \end{array}
$$
F. $$
\begin{array} {l} {{{\varphi\ ( 1 9 5 6 )=6 5 5}}} \\ {{{\sigma( 1 9 5 6 )=6 3 3 487}}} \\ {{{\varphi\ ( 5 0 4 0 )=1 1 1 0}}} \\ {{{\sigma( 5 0 4 0 )=2 9 2 5 387}}} \\ \end{array}
$$
G. $$
\begin{array} {l} {{{\varphi\ ( 1 9 5 6 )=6 3 8}}} \\ {{{\sigma( 1 9 5 6 )=6 3 3 1 93}}} \\ {{{\varphi\ ( 5 0 4 0 )=1 1 21}}} \\ {{{\sigma( 5 0 4 0 )=2 9 1 0 1 0 0}}} \\ \end{array}
$$
H. $$
\begin{array} {l} {{{\varphi\ ( 1 9 5 6 )=6 9 5}}} \\ {{{\sigma( 1 9 5 6 )=6 3 5 0 32}}} \\ {{{\varphi\ ( 5 0 4 0 )=1 1 4 1}}} \\ {{{\sigma( 5 0 4 0 )=2 9 0 4 4 5 5}}} \\ \end{array}
$$
I. $$
\begin{array} {l} {{{\varphi\ ( 1 9 5 6 )=7 02}}} \\ {{{\sigma( 1 9 5 6 )=6 3 4 5 0 1}}} \\ {{{\varphi\ ( 5 0 4 0 )=1 1 2 0}}} \\ {{{\sigma( 5 0 4 0 )=2 9 08136}}} \\ \end{array}
$$
J. $$
\begin{array} {l} {{{\varphi\ ( 1 9 5 6 )=6 2 5}}} \\ {{{\sigma( 1 9 5 6 )=6 3 5 5 5 5}}} \\ {{{\varphi\ ( 5 0 4 0 )=1 1 8 0}}} \\ {{{\sigma( 5 0 4 0 )=2 9 4 12 09}}} \\ \end{array}
$$
Answer: E
Find the solution to the congruences $\left\{\begin{matrix} {{x+4 y-2 9 \equiv0 ( \mathrm{mod} 1 4 3 )}} \\ {{2 x-9 y+8 4 \equiv0 ( \mathrm{mod} 1 4 3 )}} \\ \end{matrix} \right.$.

Options:
A. $$
\begin{array} {l} {{x \equiv4 ( \mathrm{m o d} 1 4 3 )}} \\ {{y \equiv4 2 ( \mathrm{m o d} 1 4 3 )}} \\ \end{array}
$$
B. $$
\begin{array} {l} {{x \equiv0 ( \mathrm{m o d} 1 4 3 )}} \\ {{y \equiv3 2 ( \mathrm{m o d} 1 4 3 )}} \\ \end{array}
$$
C. $$
\begin{array} {l} {{x \equiv2 ( \mathrm{m o d} 1 4 3 )}} \\ {{y \equiv4 7 ( \mathrm{m o d} 1 4 3 )}} \\ \end{array}
$$
D. $$
\begin{array} {l} {{x \equiv6 ( \mathrm{m o d} 1 4 3 )}} \\ {{y \equiv5 3 ( \mathrm{m o d} 1 4 3 )}} \\ \end{array}
$$
E. $$
\begin{array} {l} {{x \equiv7 ( \mathrm{m o d} 1 4 3 )}} \\ {{y \equiv3 6 ( \mathrm{m o d} 1 4 3 )}} \\ \end{array}
$$
F. $$
\begin{array} {l} {{x \equiv1 ( \mathrm{m o d} 1 4 3 )}} \\ {{y \equiv4 5 ( \mathrm{m o d} 1 4 3 )}} \\ \end{array}
$$
G. $$
\begin{array} {l} {{x \equiv5 ( \mathrm{m o d} 1 4 3 )}} \\ {{y \equiv3 9 ( \mathrm{m o d} 1 4 3 )}} \\ \end{array}
$$
H. $$
\begin{array} {l} {{x \equiv9 ( \mathrm{m o d} 1 4 3 )}} \\ {{y \equiv3 8 ( \mathrm{m o d} 1 4 3 )}} \\ \end{array}
$$
I. $$
\begin{array} {l} {{x \equiv8 ( \mathrm{m o d} 1 4 3 )}} \\ {{y \equiv4 1 ( \mathrm{m o d} 1 4 3 )}} \\ \end{array}
$$
J. $$
\begin{array} {l} {{x \equiv3 ( \mathrm{m o d} 1 4 3 )}} \\ {{y \equiv4 8 ( \mathrm{m o d} 1 4 3 )}} \\ \end{array}
$$
Answer: A
Find all natural numbers that satisfy $3^{m}-1$ being divisible by $2^{m}$.

Options:
A. 3,5,7
B. 1,4,5
C. 2,5,6
D. 1,3,5
E. 2,4,6
F. 1,2,4
G. 1,3,4
H. 2,3,5
I. 2,3,4
J. 1,2,3
Answer: F
Two individuals, $A$ and $B$, plan to travel from point $M$ to point $N$. It is known that the distance between $M$ and $N$ is 15 km. If walking, each person can cover $6 \mathrm{~km}$ per hour. If riding a bicycle, they can travel $15 \mathrm{~km}$ per hour. $A$ sets out from $M$ on foot, while $B$ starts from $M$ on a bicycle at the same time. When $B$ encounters a pedestrian $C$ walking from $N$ towards $M$, $B$ hands the bicycle over to $C$ and $C$ rides the bicycle towards $M$, while $B$ switches to walking towards $N$. When $C$ meets $A$, $C$ hands over the bicycle to $A$, and then $A$ rides the bicycle to $N$.When should pedestrian $C$ leave $N$, so that $C$ and $B$ arrive at $N$ simultaneously, assuming $C$'s walking speed is also $6 \mathrm{~km/h}$?

Options:
A. $C$ should depart from $N$ $\frac{3}{11} \mathrm{h}$ before $A$ and $B$ set off from $M$.
B. $C$ should depart from $N$ $\frac{4}{11} \mathrm{h}$ after $A$ and $B$ set off from $M$.
C. $C$ should depart from $N$ $\frac{5}{12} \mathrm{h}$ before $A$ and $B$ set off from $M$.
D. $C$ should depart from $N$ $\frac{3}{13} \mathrm{h}$ before $A$ and $B$ set off from $M$.
E. $C$ should depart from $N$ $\frac{7}{25} \mathrm{h}$ before $A$ and $B$ set off from $M$.
F. $C$ should depart from $N$ $\frac{2}{11} \mathrm{h}$ after $A$ and $B$ set off from $M$.
G. $C$ should depart from $N$ $\frac{1}{2} \mathrm{h}$ after $A$ and $B$ set off from $M$.
H. $C$ should depart from $N$ $\frac{9}{22} \mathrm{h}$ before $A$ and $B$ set off from $M$.
I. $C$ should depart from $N$ $\frac{2}{15} \mathrm{h}$ after $A$ and $B$ set off from $M$.
J. $C$ should depart from $N$ $\frac{1}{3} \mathrm{h}$ after $A$ and $B$ set off from $M$.
Answer: A
Calculate the value of $\begin{aligned}\sqrt[6]{7\sqrt[3]{20}-19}\end{aligned}$?

Options:
A. $\sqrt[3]{\frac67}-\sqrt{\frac23}$
B. $\sqrt[3]{\frac53}-\sqrt{\frac43}$
C. $\sqrt[3]{\frac73}-\sqrt{\frac23}$
D. $\sqrt[3]{\frac53}-\sqrt{\frac13}$
E. $\sqrt[3]{\frac43}-\sqrt{\frac23}$
F. $\sqrt[3]{\frac53}-\sqrt{\frac23}$
G. $\sqrt[3]{\frac53}+\sqrt{\frac13}$
H. $\sqrt[3]{\frac53}-\sqrt{\frac53}$
I. $\sqrt[3]{\frac53}+\sqrt{\frac23}$
J. $\sqrt[3]{\frac23}-\sqrt{\frac23}$
Answer: F
What is the value of $\begin{aligned}\sqrt[3]{\sqrt[5]{\frac{32}{5}}-\sqrt[5]{\frac{27}{5}}}\end{aligned}$?

Options:
A. $\sqrt[5]{\frac1{49}}+\sqrt[5]{\frac3{16}}-\sqrt[5]{\frac9{16}}$
B. $\sqrt[5]{\frac1{25}}+\sqrt[5]{\frac3{25}}-\sqrt[5]{\frac9{25}}$
C. $\sqrt[5]{\frac1{36}}+\sqrt[5]{\frac3{25}}-\sqrt[5]{\frac9{25}}$
D. $\sqrt[5]{\frac1{25}}+\sqrt[5]{\frac4{25}}-\sqrt[5]{\frac9{16}}$
E. $\sqrt[5]{\frac1{25}}+\sqrt[5]{\frac3{16}}-\sqrt[5]{\frac9{25}}$
F. $\sqrt[5]{\frac1{16}}+\sqrt[5]{\frac3{25}}-\sqrt[5]{\frac9{25}}$
G. $\sqrt[5]{\frac2{25}}+\sqrt[5]{\frac4{25}}-\sqrt[5]{\frac9{25}}$
H. $\sqrt[5]{\frac1{25}}+\sqrt[5]{\frac4{25}}-\sqrt[5]{\frac9{49}}$
I. $\sqrt[5]{\frac1{27}}+\sqrt[5]{\frac3{27}}-\sqrt[5]{\frac9{27}}$
J. $\sqrt[5]{\frac1{25}}+\sqrt[5]{\frac5{25}}-\sqrt[5]{\frac9{36}}$
Answer: B
When purchasing granite slabs at the quarry, there are 120 slabs each weighing 7t and 80 slabs each weighing 90t. Each railway flatcar can carry a maximum load of 40t at a time. How many railway flatcars are needed at least to transport all the slabs at once?

Options:
A. 50
B. 30
C. 55
D. 65
E. 75
F. 60
G. 40
H. 45
I. 35
J. 70
Answer: G