Question

Convert gravitational constant (G) from CGS to MKS system.

Answer 1

Hint:To solve the given question we must know the value of the gravitational constant in the CGS system. We should also know the relation between the units of the CGS and the MKS system. Substitute the values of the CGS units in terms of MKS units and find the value of G in the MKS system.

Formula used:
$1cm={10}^{-2}m$
$1g={10}^{-3}kg$.

Complete step by step answer:
Let us first understand what the gravitational constant is.
Gravitational constant is a universal proportionality constant that is used in the equation for the gravitational force between two point sized particles.

The gravitational force in the above case is given as,
$F={\displaystyle \frac{G{m}_1{m}_2}{r^2}}$, where G is the gravitational constant, $m_1$ and $m_2$ are the masses of the two particles and r is the distance between them.

The value of the gravitational constant (G) in CGS system is,
$G=6.67\times {10}^{-8}c{m}^3{g}^{-1}{s}^{-2}$ ….. (i).
Here, cm (centimetre), g (gram) and s (second) are the units of length, mass and time in the CGS system of units respectively. The MKS units of length mass and time are m (metre), kg (kilogram) and s (second) respectively.

To convert the value of the gravitational constant from CGS system into MKS system we will use the following relations:
$1cm={10}^{-2}m$
$1g={10}^{-3}kg$.
Substitute the values of dyne, cm and g in equation (i).
Therefore, we get that
$G=6.67\times {10}^{-8}{\left({10}^{-2}m\right)}^3{\left({10}^{-3}kg\right)}^{-1}{s}^{-2}$
$\Rightarrow G=6.67\times {10}^{-8}\left({10}^{-6}{m}^3\right)\left({10}^{3}k{g}^{-1}\right)s^{-2}$
This further implies that, $G=6.67\times {10}^{-11}{m}^{3}k{g}^{-1}{s}^{-2}$.

Hence, the value of the gravitational constant in MKS system is $G=6.67\times {10}^{-11}{m}^{3}k{g}^{-1}{s}^{-2}$.

Note:Note that the MKS (metre kilogram second) system and the SI (Standard International) system are the same system of units. Both the systems use metre (m), kilogram (kg) and seconds (s) as the units of length, mass and time. If you do not know the unit of G, then you can find it from the equation for the gravitational force. From the equation, you can calculate the dimensional formula of G and know the unit of G.