A hypothetical equation is given as $\rho = a \sqrt{P} + bT^2$, where $\rho$ is density, $P$ is pressure, and $T$ is temperature. The dimensions of $\frac{a}{b}$ are:

The viscous force $F$ acting on a sphere moving through a liquid depends on the radius $r$ of the sphere, the velocity $v$ of the sphere, and the coefficient of viscosity $η$ of the liquid. If $F = kη^arv^b$, where $k$ is a dimensionless constant, using dimensional analysis, find the values of $a$, $b$.

If the units of M and L are increased three times, then the unit of energy will be increased by

If ${u_1}\,and\,{u_2}$ are the units selected in two systems of measurement and ${n_1}\,and\,{n_2}$ their numerical values, then

The SI unit of length is the metre. Suppose we adopt a new unit of length which equal x metre. The area of $1{m^2}$ expressed in terms of the new unit has a magnitude

Which does not have the same unit as others

The period $T$ of oscillation of a spring-mass system depends on the mass $m$ and the spring constant $k$. Using dimensional analysis and the principle of homogeneity, determine the relationship between $T$, $m$, and $k$.

If 1 g cm ${{\rm{s}}^{ - 1}} = x$ newton-second, then the number x is equal to

If the units of mass, length and time are doubled, unit of angular momentum will be

Which relation is wrong

The time period of a simple pendulum is given by $T = 2\pi\sqrt{\frac{l}{g}} + kx$, where $l$ is the length, $g$ is acceleration due to gravity, and $x$ is displacement. The dimensions of $k$ are: