Consider a fluid in a container. Let the density of the fluid at the surface and at depth $H$ be $\rho_0$ and $\rho$ respectively. The bulk modulus of the fluid is $B_0$. If $\rho = \frac{\rho_0}{1 - \frac{\rho g H}{B_0}}$, find the constant $\alpha$ in the expression $\rho = \frac{\rho_0}{1 + \alpha \rho g H}$ (Assume $\frac{\rho - \rho_0}{\rho_0} \ll 1$).

The compressibility of water is $4 \times 10^{-5}$ per unit atmospheric pressure. The decrease in volume of $100 \ cm^3$ of water under a pressure of $100 \ atm$ will be ......... $cm^3$.

$A$ glass slab is subjected to a pressure of $10 \, atm$. The fractional change in its volume is (Bulk modulus of glass $= 37 \times 10^9 \, N \, m^{-2}$,$1 \, atm = 1 \times 10^5 \, N \, m^{-2}$)

Calculate the stress developed inside a tooth cavity filled with copper when hot tea at a temperature of $57\,^{\circ}C$ is drunk. You can take the body (tooth) temperature to be $37\,^{\circ}C$ and $\alpha = 1.7 \times 10^{-5}/^{\circ}C$,bulk modulus for copper $= 140 \times 10^9\, N/m^2$.

$A$ 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 the entire cross-section of the 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:

Which of the following moduli of elasticity is possible in solids,liquids,and gases?