$A$ $U$-tube of uniform bore of cross-sectional area $A$ has been set up vertically with open ends facing up. Now $M$ grams of a liquid of density $d$ is poured into it. The column of liquid in this tube will oscillate with a period $T$ such that

$A$ circular disc of mass $10 \; kg$ is suspended by a wire attached to its centre. The wire is twisted by rotating the disc and released. The period of torsional oscillations is found to be $1.5 \; s$. The radius of the disc is $15 \; cm$. Determine the torsional spring constant of the wire in $N \; m \; rad^{-1}$. (Torsional spring constant $\alpha$ is defined by the relation $J = -\alpha \theta$,where $J$ is the restoring couple and $\theta$ is the angle of twist).

$A$ bead of mass $m$ is attached to the mid-point of a taut,weightless string of length $l$ and placed on a frictionless horizontal table. Under a small transverse displacement $x$,as shown in the figure,if the tension in the string is $T$,then the frequency of oscillation is:

$A$ sphere of radius '$r$' is kept on a concave mirror of radius of curvature '$R$'. The arrangement is kept on a horizontal table. If the sphere is displaced from its equilibrium position and left,then it executes $S$.$H$.$M$. The period of oscillation will be ($g =$ acceleration due to gravity).

$9 \text{ kg}$ of solution is poured into a glass $U$-tube as shown in the figure. The tube's inner diameter is $2 \sqrt{\frac{\pi}{5}} \text{ m}$ and the solution oscillates freely up and down about its position of equilibrium $(x=0)$. The period of oscillation in seconds is (Given: $1 \text{ m}^3$ of solution has a mass $\rho=900 \text{ kg/m}^3$, $g=10 \text{ m/s}^2$, ignore frictional and surface tension effects).

$9 \ kg$ 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 $\rho = 13.6 \times 10^3 \ kg/m^3$)