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 unit of length and force be increased four times, then the unit of energy is

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 $a$, $b$.

The unit of potential energy is

If the velocity ($v$), acceleration ($a$), and time ($t$) are related by an equation $v = at + bt^2$, what are the dimensions of $b$?

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?

Which of the following units denotes the dimensions $\left[ {{\rm{M}}{{\rm{L}}^2}/{{\rm{Q}}^2}} \right],$, where Q denotes the electric charge?

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 $k$.

Consider a new system of units in which c(speed of light in vacuum), h(Planck’s constant) and G(gravitational constant) are taken as fundamental units. Which of the following would correctly represent mass in this new system?

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?

Given the equation $P = \frac{a}{V} - \frac{b}{V^2} + cT$, where $P$ is pressure, $V$ is volume, and $T$ is temperature. What are the dimensions of $abc$?