20. A spectrometer with a grating spacing \(d_{\lambda}\) is used to separate two laser beams of wavelengths \(\lambda\) and \(\lambda + \Delta \lambda\) (\(\Delta \lambda << \lambda\)) by the 2nd order diffraction. The difference of the diffraction angles \(\Delta \theta\) is

(A)\(\frac{2\Delta\lambda}{d\cos\theta}\)
(B)\(\frac{\Delta\lambda\sin\theta}{2d}\)
(C)\(\frac{\Delta\lambda\tan\theta}{2d}\)
(D)\(\frac{4\Delta\lambda}{d\sin\theta}\)

(E)None of the above

1. Assuming that you accelerate your car from $10\mathrm{m/s}$ to $25\mathrm{m/s}$ in a distance of $87.5\mathrm{m}$ with a constant acceleration, then the force your body exerts on the car seat (let your mass be $75\mathrm{kg}$) is(A)$550\mathrm{N}$(B)$1100\mathrm{N}$(C)$225\mathrm{N}$(D)$112.5\mathrm{N}$(E)$337.5\mathrm{N}$

2. The initial temperature and volume of an ideal gas are $127^{\circ}\mathrm{C}$ and $23.6\mathrm{L}$ respectively. After an adiabatic expansion, the final temperature and volume of the gas are $27^{\circ}\mathrm{C}$ and $56\mathrm{L}$ respectively. The constant-volume and constant-pressure heat capacities of this ideal gas are(A)$C_V = 1.5R, C_P = 2.5R$(B)$C_V = 2.0R, C_P = 3.0R$(C)$C_V = 2.5R, C_P = 3.5R$(D)$C_V = 3.0R, C_P = 4.0R$(E)$C_V = 3.5R, C_P = 4.5R$

3. The molecular bonding in a diatomic molecule such as the nitrogen $(\mathbf{N}_2)$ molecule can be modeled by the Lennard-Jones potential, which has the form $U(x) = 4U_0\left(\left(\frac{x_0}{x}\right)^{12} - \left(\frac{x_0}{x}\right)^6\right)$, where $x$ is the separation distance between the two nuclei and $x_0$, and $U_0$ are constants. The equilibrium separation $x_0$, which is the value of $x$ for which the two atoms experience zero force from each other, is equal to(A)$2^{-1/6}x_0$(B)$3^{1/6}x_0$(C)$3^{1/3}x_0$(D)$3^{-1/6}x_0$(E)$2^{1/6}x_0$

4. Let us suppose that $n$ mole of ideal gas has gone through a free expansion $(T_i = T_f)$ from an initial volume $V_i$ to the final volume $V_f$. The entropy change in such a process can be expressed as a function $\Delta S(n, V_i, V_f)$. One has $\Delta S(3, 200\mathrm{cm}^3, 600\mathrm{cm}^3)$ closest to (A)4.50 R(B)1.65 R(C)2.25 R(D)3.30 R(E)6.60 R

5. A liquid of density $900\mathrm{kg/m^3}$ flows through a horizontal pipe that has a cross-sectional area of $1.90\times 10^{-2}\mathrm{m^2}$ in region A and a cross-sectional area of $9.50\times 10^{-2}\mathrm{m^2}$ in region B. The pressure difference between two regions is $7.20\times 10^3$ Pa. The volume flow rate $R_V$ of the liquid is(A)$0.776\mathrm{m^3/s}$(B)$0.0776\mathrm{m^3/s}$(C)$1.552\mathrm{m^3/s}$(D)$0.1552\mathrm{m^3/s}$(E)$0.3104\mathrm{m^3/s}$

6. The mass and radius of a spherical planet with a uniform density are M and R respectively. The planet constantly rotates with a period T. The free-fall acceleration $g_{p}$ at the pole of the planet (where the rotation axis passes through) and the free-fall acceleration $g_{e}$ at the equator of the planet are related to each other by(A)$g_{e} - g_{p} = \frac{4\pi^{2}R}{T^{2}}$(B)$g_{e} - g_{p} = \frac{2\pi^{2}R}{T^{2}}$(C)$g_{e} - g_{p} = -\frac{2\pi^{2}R}{T^{2}}$(D)$g_{e} - g_{p} = -\frac{4\pi^{2}R}{T^{2}}$(E)None of the above

7. One end of a $2.0\mathrm{m}$ long uniform meter stick is initially held at rest at $\theta = 40^{\circ}$ while the other end of the stick is on the floor. The meter stick is then released. Assuming the end on the floor does not slip, then the angular speed of the stick at the time when the stick is about to hit the floor is closest to (A)3.1 rad/s(B)6.2 rad/s(C)4.5 rad/s(D)9.0 rad/s(E)13.5 rad/s

8. A rigid body rotates at 300 revolutions per minute around an axis which goes through the origin and points into the first octant, making an angle of $60^{\circ}$ with both the x- and y-axis. The velocity of a point $\vec{r} = (0,0,2)\mathrm{cm}$ is (A)$\vec{v} = (0,0,600)\mathrm{cm/s}$(B)$\vec{v} = (0,19.7,14.8)\mathrm{cm/s}$(C)$\vec{v} = (-31.4,31.4,0)\mathrm{cm/s}$(D)$\vec{v} = (31.4,31.4,0)\mathrm{cm/s}$(E)$\vec{v} = (31.4,-31.4,0)\mathrm{cm/s}$

9. The speed of sound in oxygen, at $0^{\circ}\mathrm{C}$, is $317\mathrm{m/s}$. At the same temperature and pressure, the speed of sound in hydrogen, in $\mathrm{m/s}$, is closest to which one of the following?(A)79(B)159(C)634(D)317(E)1268

10. Suppose that a simple pendulum consists of a small $60.0\mathrm{g}$ at the end of a cord of negligible mass. If the angle $\theta$ between the cord and the vertical is given by $\theta = (0.080\mathrm{rad})\cos[(4.43\mathrm{rad/s})x + \phi]$, then the pendulum's length is(A)$314\mathrm{cm}$(B)$50\mathrm{cm}$(C)$100\mathrm{cm}$(D)$157\mathrm{cm}$(E)$628\mathrm{cm}$

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