A solid sphere of radius $r$ made of a soft material of bulk modulus $K$ is surrounded by a liquid in a cylindrical container. A massless piston of area a floats on the surface of the liquid, covering entire cross-section of cylindrical container. When a mass m is placed on the surface of the piston to compress the liquid, the fractional decrement in the radius of the sphere$\left( {\frac{{dr}}{r}} \right)$ is

Two small drops of mercury, each of radius $R$ coalesce to form a single large drop. The radio of the total surface energies before and after the change is

A wooden cube first floats inside water when a $200\,g$ mass is placed on it. When the mass is removed the cube is $2\,cm$ above water level. The side of cube is ……… $cm$

A spherical body of mass $m$ and radius $r$ is allowed to fall in a medium of viscosity $\eta $. The time in which the velocity of the body increases from zero to $0.63\, times$ the terminal velocity $(v)$ is called time constant $\left( \tau  \right)$. Dimensionally $\tau $ can be represented by

If the terminal speed of a sphere of gold (density $19.5 \,kg / m ^2$ ) is $0.2 \,m / s$ in a viscous liquid (density $=1.5 \,kg / m ^3$ ), find the terminal speed of a sphere of silver (density $=10.5 \,kg / m ^3$ ) of the same size in the same liquid is ………… $m / s$