Consider a spongy block of mass $m$ floating on a flowing river. The maximum mass of the block is related to the speed of the river flow $v$,acceleration due to gravity $g$,and the density of the block $\rho$ such that $m_{\max} = k v^x g^y \rho^z$ ($k$ is a constant). The values of $x, y$,and $z$ should then respectively be:
(Mass of the spongy block is assumed to vary due to absorption of water by it)

The equation of state of some gases can be expressed as $(P + \frac{a}{V^2}) = \frac{b\theta}{l}$,where $P$ is the pressure,$V$ is the volume,$\theta$ is the absolute temperature,and $a$ and $b$ are constants. The dimensional formula of $a$ is

The time period of a body undergoing simple harmonic motion is given by $T=p^{a} D^{b} S^{c}$,where $p$ is the pressure,$D$ is density,and $S$ is surface tension. The values of $a, b,$ and $c$ respectively are

$A$ length-scale $(l)$ depends on the permittivity $(\varepsilon)$ of a dielectric material,the Boltzmann constant $(k_B)$,the absolute temperature $(T)$,the number density $(n)$ of certain charged particles,and the charge $(q)$ carried by each of the particles. Which of the following expression$(s)$ for $l$ is(are) dimensionally correct?
$(A)$ $l=\sqrt{\left(\frac{n q^2}{\varepsilon k_B T}\right)}$
$(B)$ $l=\sqrt{\left(\frac{\varepsilon k_B T}{n q^2}\right)}$
$(C)$ $l=\sqrt{\left(\frac{q^2}{\varepsilon n^{2 / 3} k_B T}\right)}$
$(D)$ $l=\sqrt{\left(\frac{q^2}{\varepsilon n^{1 / 3} k_B T}\right)}$

The frequency $(v)$ of an oscillating liquid drop may depend upon the radius $(r)$ of the drop,the density $(\rho)$ of the liquid,and the surface tension $(s)$ of the liquid as: $v = r^{a} \rho^{b} s^{c}$. The values of $a, b,$ and $c$ respectively are:

The potential energy of a particle varies with distance $x$ from a fixed origin as $U = \frac{A\sqrt{x}}{x^2 + B}$,where $A$ and $B$ are dimensional constants. Find the dimensional formula for $A/B$.