If $E, M, J$ and $G$ respectively denote energy, mass, angular momentum, and universal gravitational constant, the quantity which has the same dimensions as the dimensions of $\frac{E J^2}{M^5 G^2}$ is:

The $SI$ unit of energy is $J = kg \, m^{2} \, s^{-2}$; that of speed $v$ is $m \, s^{-1}$ and of acceleration $a$ is $m \, s^{-2}$. Which of the formulae for kinetic energy $(K)$ given below can you rule out on the basis of dimensional arguments ($m$ stands for the mass of the body):
$(a)$ $K = m^{2} v^{3}$
$(b)$ $K = (1/2) m v^{2}$
$(c)$ $K = m a$
$(d)$ $K = (3/16) m v^{2}$
$(e)$ $K = (1/2) m v^{2} + m a$

$A$ wave pulse can travel along a tense string like a violin string. $A$ series of experiments showed that the wave velocity $V$ of a pulse depends on the following quantities: the tension $T$ of the string,the cross-sectional area $A$ of the string,and the density $\rho$ (mass per unit volume) of the string. Obtain an expression for $V$ in terms of $T$,$A$,and $\rho$ using dimensional analysis.

$A$ calorie is a unit of heat or energy and it equals about $4.2 \; J$ where $1 \; J = 1 \; kg \; m^2 \; s^{-2}$. Suppose we employ a system of units in which the unit of mass equals $\alpha \; kg$, the unit of length equals $\beta \; m$, and the unit of time equals $\gamma \; s$. Show that a calorie has a magnitude $4.2 \; \alpha^{-1} \beta^{-2} \gamma^2$ in terms of the new units.

Let force $F = A \sin(Ct) + B \cos(Dx)$,where $x$ and $t$ are displacement and time respectively. The dimensions of $\frac{C}{D}$ are the same as the dimensions of

If electronic charge $e$,electron mass $m$,speed of light in vacuum $c$,and Planck's constant $h$ are taken as fundamental quantities,the permeability of vacuum $\mu_0$ can be expressed in units of