In both figures shown below, there is a hole along the diameter of the Earth. In the first figure, a particle is released from point $A$ and it oscillates with a time period $T_1$. In the second figure, the same particle is released from point $B$ and it oscillates with a time period $T_2$. Then, [$O$ is the centre of the Earth]

$A$ particle is executing a simple harmonic motion. Its maximum acceleration is $\alpha$ and maximum velocity is $\beta$. Then,its time period of vibration will be

$9 \ kg$ of mercury is poured into a glass $U$-tube with an inner diameter of $1.2 \ cm$. The mercury can flow without friction within the tube. The oscillation period is ......... $\sec$. (Density of mercury $\rho = 13.6 \times 10^3 \ kg/m^3$)

One end of a $V$-tube containing mercury is connected to a suction pump and the other end to the atmosphere. The two arms of the tube are inclined to the horizontal at an angle of $45^{\circ}$ each. $A$ small pressure difference is created between the two columns when the suction pump is removed. Will the column of mercury in the $V$-tube execute simple harmonic motion? Neglect capillary and viscous forces. Find the time period of oscillation.

$A$ large horizontal surface moves up and down in $S.H.M.$ with an amplitude of $1 \, cm$. If a mass of $10 \, kg$ placed on the surface is to remain continuously in contact with it,the maximum frequency of $S.H.M.$ will be .... $Hz$.

$A$ uniform rod of length $L$ and mass $M$ is pivoted at the centre. Its two ends are attached to two springs of equal spring constants $k$. The springs are fixed to rigid supports as shown in the figure,and the rod is free to oscillate in the horizontal plane. The rod is gently pushed through a small angle $\theta$ in one direction and released. The frequency of oscillation is