A cylindrical piece of cork of density of base area A and height h floats in a liequid of density p_(1) . The cork is depressed slightly and then released. Show that the cork oscillates up and down simple harmonically with a period. T= 2pi sqrt((hp)/(p_(1)g) Where p is the density of cork. (Ignore damping due to viscosity of the liquid).

A cylindrical of wood (density = 600 kg m^(-3) ) of base area 30 cm^(2) and height 54 cm , floats in a liquid of density 900 kg^(-3) The block is displaced slightly and then released. The time period of the resulting oscillations of the block would be equal to that of a simple pendulum of length (nearly) :

A cylindrical log of wood of height h and area of cross-section A floats in water. It is pressed and then released. Show that the log would execute SHM with a time period. T= 2pi sqrt((m)/(A p g)) where, m is mass of the body and p is density of the liquid. .

A wooden cube (density of wood 'd' ) of side 'l' flotes in a liquid of density 'rho' with its upper and lower surfaces horizonta. If the cube is pushed slightly down and released, it performs simple harmonic motion of period 'T' . Then, 'T' is equal to :-

Two identical cylindrical vessels with their bases at the same level each, contain a liquid of density rho . The area of either base is A but in one vessel the liquid heigth is h_(1) and in the other liquid height is h_(2)(h_(2)lth_(1)) . If the two vessel are connected, the work done by gravity in equalizing the level is

A cylinderical rod of length l=2m & density (rho)/(2) floats vertically in a liquid of density rho as shown in figure (i). Show that it performs SHM when pulled slightly up & released find its time period. Neglect change in liquid level. (ii). Find the time taken by the rod to completely immerse when released from position shown in (b). Assume that it remains vertical throughout its motion (take g=pi^(2)m//s^(2) )

A small spherical monoatomic ideal gas bubble (gamma= (5)/(3)) is trapped inside a liquid of density rho_(l) (see figure) . Assume that the bubble does not exchange any heat with the liquid. The bubble contains n moles of gas. The temperature of the gas when the bubble is at the bottom is T_(0) , the height of the liquid is H and the atmospheric pressure is P_(0) (Neglect surface tension). When the gas bubble is at a height y from the bottom , its temperature is :-

A homogeneous solid cylinder of length L(LltH/2), cross-sectional area A/5 is immersed such that it floats with its axis vertical at the liquid-liquid interface with length L/4 in the denser liquid as shown in the figure. The lower density liquid is open to atmosphere having pressure P_0 . Then density D of solid is given by

Two spheres P and Q of equal radii have densities rho_1 and rho_2 , respectively. The spheres are connected by a massless string and placed in liquids L_1 and L_2 of densities sigma_1 and sigma_2 and viscosities eta_1 and eta_2 , respectively. They float in equilibrium with the sphere P in L_1 and sphere Q in L_2 and the string being taut(see figure). If sphere P alone in L_2 has terminal velocity vecV_p and Q alone in L_1 has terminal velocity vecV_Q , then

A container of large uniform cross-sectional area A resting on a horizontal surface, holes two immiscible, non-viscon and incompressible liquids of densities d and 2d each of height H//2 as shown in the figure. The lower density liquid is open to the atmosphere having pressure P_(0) . A homogeneous solid cylinder of length L(LltH//2) and cross-sectional area A//5 is immeresed such that it floats with its axis vertical at the liquid-liquid interface with length L//4 in the denser liquid, The cylinder is then removed and the original arrangement is restroed. a tiny hole of area s(sltltA) is punched on the vertical side of the container at a height h(hltH//2) . As a result of this, liquid starts flowing out of the hole with a range x on the horizontal surface. The total pressure with cylinder, at the bottom of the container is