6. If the mass density of the Earth is \( \rho \), for a tunnel drilled through between the north and south poles, what is the period of oscillation of a particle pulled by the gravity and moving without friction in the tunnel from one pole to the other?
(A)\( \sqrt{\frac{3\pi}{G\rho}} \)
(B)\( \sqrt{\frac{3\pi}{4G\rho}} \)
(C)\( \sqrt{\frac{3}{4\pi G\rho}} \)
(D)\( \sqrt{\frac{3\pi}{2G\rho}} \)

(E)\( \sqrt{\frac{4\pi}{G\rho}} \)

1. Four particles are moving on a plane under the same constant acceleration \( \vec{a} = \hat{y} \), each having its initial velocity vector as follows: (a)\( 10\hat{x}+10\hat{y} \), (b)\( 10\hat{x}+8\hat{y} \), (c)\( 8\hat{x}+10\hat{y} \), (d)\( 12\hat{x}+12\hat{y} \). If the respective displacements are \( R_a \), \( R_b \), \( R_c \), \( R_d \), which of the following is correct?(A)\( R_a ＞ R_b ＞ R_c ＞ R_d \)(B)\( R_c ＞ R_b ＞ R_d ＞ R_a \)(C)\( R_d ＞ R_a ＞ R_b ＞ R_c \)(D)\( R_d ＞ R_a ＞ R_c ＞ R_b \)(E)\( R_a ＞ R_c ＞ R_b ＞ R_d \)

2. Given a position vector \( \vec{r} = (-t^2+1)\hat{i} + (3t-5)\hat{j} + (t^2-2t+2)\hat{k} \), find the magnitude of acceleration at t=1.(A)\( \sqrt{5} \)(B)\( \sqrt{14} \)(C)\( 2\sqrt{2} \)(D)2(E)0

3. What is the rotational inertia of a solid sphere about an axis that passes through the spherical center? (Assume the radius is R and mass is M.)(A)\( \frac{1}{2}MR^2 \)(B)\( \frac{2}{3}MR^2 \)(C)\( \frac{1}{4}MR^2 \)(D)\( \frac{2}{5}MR^2 \)(E)\( MR^2 \)

4. A man of mass m falls from rest at point A that is at a horizontal distance of D from the origin O of an xyz coordinate system, i.e., A = (D, 0, 0). Assuming gravity \( \vec{g} = -g\hat{z} \). What is the angular momentum of the falling man about O at time t?(A)\( Dmgt\hat{x} \)(B)\( \frac{1}{2}Dmgt\hat{x} \)(C)\( Dmgt\hat{y} \)(D)\( \frac{1}{2}Dmgt\hat{y} \)(E)\( -\frac{1}{2}Dmgt\hat{z} \)

5. Which wheel is at equilibrium?(A)(B)(C)(D)(E)E

7. A linear SHM occurs at a frequency of f=2Hz. If initially at t=0 the displacement x(t) is x(0)=3, and the velocity v(0)=0, what is v(0.5)?(A)0(B)\( \pi \)(C)\( 2\pi \)(D)\( 3\pi \)(E)\( 6\pi \)

8. A sound source of frequency f moves at a velocity of \( v_s \), while a detector moves at \( v_d \) in parallel. If the speed of the sound in air is v, what is the frequency of the sound as measured by the detector?(A)\( \frac{v+v_d}{v+v_s}f \)(B)\( \frac{v-v_d}{v-v_s}f \)(C)\( \frac{v+v_s}{v+v_d}f \)(D)\( \frac{v-v_d}{v+v_s}f \)(E)\( \frac{v+v_s}{v-v_d}f \)

9. What is the heat capacity of two connected metals, each with masses \( M_1 \) and \( M_2 \) and specific heats of \( X_1 \) and \( X_2 \)?(A)\( \frac{X_1}{M_1} + \frac{X_2}{M_2} \)(B)\( \frac{M_1}{X_1} + \frac{M_2}{X_2} \)(C)\( M_1X_1 + M_2X_2 \)(D)\( \frac{X_1+X_2}{M_1+M_2} \)(E)\( \frac{M_1+M_2}{X_1+X_2} \)

10. For a particle of temperature T located in an environment \( T_{env} \), what is the temperature dependence of thermal radiation energy that is emitted from the particle?(A)\( T^4 \)(B)\( T^4_{env} \)(C)\( T^4 - T^4_{env} \)(D)\( T \)(E)\( T_{env} \)

11. The speed of molecules in an ideal gas follows Maxwell's distribution law. Which of the following is correct when comparing the average speed \( v_a \), most probable speed \( v_p \), and root mean square speed \( v_{rms} \)?(A)\( v_p ＞ v_a ＞ v_{rms} \)(B)\( v_{rms} ＞ v_a ＞ v_p \)(C)\( v_{rms} ＞ v_p ＞ v_a \)(D)\( v_a ＞ v_{rms} ＞ v_p \)(E)\( v_p ＞ v_{rms} ＞ v_a \)

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