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:

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 $a$, $b$.

Let L denote the self-inductance of coil which is in series with a capacitor of capacitance C. Which of the following has the unit second?

The unit of potential energy is

If K denotes coefficient of thermal conductivity, d the density and c the specific heat, the unit of X, where X=K/dc will be

The time period ($T$) of a simple pendulum depends on its length ($l$) and acceleration due to gravity ($g$). Which of the following is dimensionally consistent with the principle of homogeneity?

If the speed $v$ of transverse waves on a stretched string depends on the tension $T$ in the string and the mass per unit length $μ$ of the string, which of the following is dimensionally consistent?

The viscous force $F$ acting on a sphere of radius $r$ moving with velocity $v$ through a fluid of viscosity $\eta$ is given by $F = 6\pi \eta rv + kr^2v^2$. The dimensions of $k$ are:

The velocity of a wave is given by $v = \sqrt{\frac{T}{\mu}} + ax^2$, where $T$ is tension, $\mu$ is linear density, and $x$ is displacement. What are the dimensions of $a$?

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

The frequency $ν$ of vibration of a stretched string depends on its length $l$, its linear mass density $μ$, and its tension $T$. Which of the following is a possible formula for $ν$, consistent with the principle of homogeneity?