A cylindrical log of wood of height h and area of cross-section A floasts in water. It is pressed and then released. Show that the log would execute S.H.M. with a time period
`T = 2pi sqrt(m//Arhog)` where m is mass of the body and `rho` is density of the liquid.

A cylindrical piece of cork of density of base area Aand height h floats in a liquid of density rho_l . The cork is depressed slightly and then released. Show that the cork oscillates up and down simple harmonically with a period T= 2pisqrt((hrho)/(rho_1g)) where p is the density of cork. (Ignore damping due to viscosity of the liquid)

Water in U-tube executes S.H.M. Will the time period for mercury filled up to the same height in the U-tube be lesser or greater than that in case of water?

Consider a rectangular block of wood moving with a velocity v_(0) in a gas at temperature T and mass density rho . Assume the velocity is along X-axis and area of cross section of the block perpendicular to v_(0) is A. Show that the drag force on the block is 4(rho) A v_(0) (sqrt(KT))/(m) , where m is the mass of the gas molecule.

A hemi-spherical tank of radius 2 m is initially full of water and has an outlet of 12c m^2 cross-sectional area at the bottom. The outlet is opened at some instant. The flow through the outlet is according to the law v(t)=0.6sqrt(2gh(t)), where v(t) and h(t) are, respectively, the velocity of the flow through the outlet and the height of water level above the outlet and the height of water level above the outlet at time t , and g is the acceleration due to gravity. Find the time it takes to empty the tank.

An air chamber of volume V has a neck area of cross section a into which a ball of mass mjust fits and can move up and down without any friction (Fig.14.27). Show that when the ball is pressed down a little and released , it executes SHM. Obtain an expression for the time period of oscillations assuming pressure-volume variations of air to be isothermal [see Fig. 14.27

The de-Broglie wavelength of a photon is the same as the wavelength of electron, show that kinetic energy of photon is (2mclambda)/h times the kinetic energy of electron, where m is the mass of electron and c is the velocity of light.

A plane is in level flight at constant speed and each of its two wings has an area of 25 m^2 . If the speed of the air is 180 km//h over the lower wing and 234 km//h over the upper wing surface, determine the plane’s mass. (Take air density to be 1 kg m^-3 )

Find the time required for a cylindrical tank of radius r and height H to empty through a round hole of area a at the bottom. The flow through the hole is according to the law v(t)=ksqrt(2gh(t)) , where v(t) and h(t) , are respectively, the velocity of flow through the hole and the height of the water level above the hole at time t , and g is the acceleration due to gravity.

A current of 1 A flows through a wire of length 0.24m and area of cross-section 1.2mm^2 , when it is conneted to a battery of 3V. Find the number density of free electrons in the wire, if the electron mobility is 4.8 xx 10^-6 m^2 V^-1s^-1 , given that charge on electron = 1.6 xx 10^-19C