Question

If E, m, J, G denotes the energy, mass, angular momentum and gravitational constant respectively. Then the dimensions of $\dfrac{{E{J^2}}}{{{m^5}{G^2}}}$ are same as that of
A. angle
B. length
C. mass
D. time

Answer 1

Hint
The dimension of $\dfrac{{E{J^2}}}{{{m^5}{G^2}}}$ is checked by substitute dimension of each term and compare wire each option dimensions.
Dimensions of angle = ${M^0}{L^0}{T^0}$
Dimension of length = ${M^0}{L^1}{T^0}$
Dimensions of mass = ${M^1}{L^0}{T^0}$
Dimensions of time = ${M^0}{L^0}{T^1}$

Step By Step Solution
Dimensions of Energy (E) =${M^1}{L^2}{T^{ - 2}}$
Dimension of angular momentum (J) = ${M^1}{L^2}{T^{ - 1}}$
Dimensions of mass (m) = ${M^1}{L^0}{T^0}$
Dimensions of gravitational constant (G) = ${M^{ - 1}}{L^3}{T^{ - 2}}$
Put in above expression to get -
$ = \dfrac{{{M^1}{L^2}{T^{ - 2}}{{({M^1}{L^2}{T^{ - 1}})}^2}}}{{{{({M^1}{L^0}{T^0})}^5}{{({M^{ - 1}}{L^3}{T^{ - 2}})}^2}}}$
Expand above expression we get
$ = \dfrac{{{M^3}{L^6}{T^{ - 4}}}}{{{M^3}{L^6}{T^{ - 4}}}}$
$ = {M^0}{L^0}{T^0}$
This represents the dimension of angle.
Correct option: (A) angle.

Note
The dimension of G can be calculated by formula
$F = \dfrac{{G{m_1}{m_2}}}{{{r^2}}}$
By interchanging terms we get
$G = \dfrac{{F{r^2}}}{{{m_1}{m_2}}}$
Put the dimension of force = ${M^1}{L^1}{T^{ - 2}}$
Dimensions of mass and length to get dimensions of G = ${M^{ - 1}}{L^3}{T^{ - 2}}$ .