Question

What is the ratio of the SI unit of G (Gravitational Constant.) to its CGS unit?
A. 100:1
B. 1000:1
C. 10:1
D. 10000:1

Answer 1

Hint: We know that the SI Unit of G as $\dfrac{{N.{m^2}}}{{k{g^2}}}$ and the CGS unit of G as $\dfrac{{dyne.c{m^2}}}{{g{m^2}}}$. To compute the ratio divide SI unit of G with CGS unit of G.

Complete step by step answer:

We know that the S.I. unit of the gravitational constant $G = \dfrac{{N.{m^2}}}{{k{g^2}}}$
And, the CGS unit of the gravitational constant $G = \dfrac{{dyne.c{m^2}}}{{g{m^2}}}$
Therefore the ratio of the SI unit of G to its CGS unit is
$ \Rightarrow R = \dfrac{{\dfrac{{N.{m^2}}}{{k{g^2}}}}}{{\dfrac{{dyne.c{m^2}}}{{g{m^2}}}}}$
Bringing the denominator to the numerator, we get,
$ \Rightarrow R = \dfrac{{N.{m^2}}}{{k{g^2}}} \times \dfrac{{g{m^2}}}{{dyne.c{m^2}}}$
Bringing the units of force, mass, and length together, we get,
$ \Rightarrow R = \dfrac{N}{{dyne}} \times \dfrac{{{m^2}}}{{c{m^2}}} \times \dfrac{{g{m^2}}}{{k{g^2}}}$
Converting all units to CGS and substituting $1N = {10^5}dyne$, $1m = {10^2}cm$ and $1kg = {10^3}gm$ in the above equation, and cancelling the units, we get,
$ \Rightarrow R = {10^5} \times {({10^2})^2} \times \dfrac{1}{{{{\left( {{{10}^3}} \right)}^2}}}$
Simplifying the exponents of 10, using exponents are multiplied in brackets and exponent signs change bringing from denominator to numerator, we get,
$ \Rightarrow R = {10^5} \times {10^4} \times {10^{ - 6}}$
Simplifying the exponents further, as exponents are added when the same base terms are multiplied, we get,
$ \Rightarrow R = {10^{5 + 4 - 6}} = {10^3}$
Hence, the ratio of the SI unit of G (Gravitational Constant.) to its CGS unit is 1000:1.

Note: The same problem can be approached via another method if the student remembers the dimension of the gravitational constant G,
$ \Rightarrow \dim (G) = {M^{ - 1}}{L^3}{T^{ - 2}}$
We know that –
for mass M: SI unit = kg, CGS unit = gm
for length L: SI unit = m, CGS unit = cm
for time T: SI unit = s, CGS unit = s;
Therefore, the ratio between SI unit of G to the cgs unit of G is
$ \Rightarrow R = \dfrac{{k{g^{ - 1}}{m^3}{s^{ - 2}}}}{{g{m^{ - 1}}c{m^3}{s^{ - 2}}}}$
Substituting, $kg = {10^3}gm$ and $m = {10^2}cm$ and cancelling the units we get,
$ \Rightarrow R = {\left( {{{10}^3}} \right)^{ - 1}}{\left( {{{10}^2}} \right)^3}$
Simplifying the exponents in the above, we obtain the required ratio,
$ \Rightarrow R = {10^{ - 3}} \times {10^6} = {10^{ - 3 + 6}} = {10^3}$
Hence, the ratio of the SI unit of G (Gravitational Constant.) to its CGS unit is 1000:1.