Consider two physical quantities A and B related to each other as $E=\frac{B-x^2}{A t}$ where $E, x$ and $t$ have dimensions of energy, length and time respectively. The dimension of $A B$ is
$\mathrm{L}^{-2} \mathrm{M}^1 \mathrm{~T}^0$
$\mathrm{L}^2 \mathrm{M}^{-1} \mathrm{~T}^1$
$\mathrm{L}^{-2} \mathrm{M}^{-1} \mathrm{~T}^1$
$\mathrm{L}^0 \mathrm{M}^{-1} \mathrm{~T}^1$
Which of the following quantities has a unit but dimensionless?
In a particular system of units, a physical quantity can be expressed in terms of the electric charge $c$, electron mass $m_c$, Planck's constant $h$, and Coulomb's constant $k=\frac{1}{4 \pi \epsilon_0}$, where $\epsilon_0$ is the permittivity of vacuum. In terms of these physical constants, the dimension of the magnetic field is $[B]=[c]^\alpha\left[m_c\right]^\beta[h]^\gamma[k]^\delta$. The value of $\alpha+\beta+\gamma+\delta$ is. . . . .
A function $f(\theta )$ is defined as $f(\theta )\, = \,1\, - \theta + \frac{{{\theta ^2}}}{{2!}} - \frac{{{\theta ^3}}}{{3!}} + \frac{{{\theta ^4}}}{{4!}} + ...$ Why is it necessary for $f(\theta )$ to be a dimensionless quantity ?
If dimensions of critical velocity $v_c$ of a liquid flowing through a tube are expressed as$ [\eta ^x \rho ^yr^z]$ where $\eta ,\rho $ and $r $ are the coefficient of viscosity of liquid, density of liquid and radius of the tube respectively, then the values of $x, y$ and $z$ are given by