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Question: If the value of G, the gravitational constant in the CGS system is \(6.7\times {{10}^{-8}}\) , then ...

If the value of G, the gravitational constant in the CGS system is 6.7×10−86.7\times {{10}^{-8}} , then what is its value in S.I. unit?

Explanation

Solution

Use the equation given for the gravitational force between two bodies and find an expression for the gravitational constant (G). Then find the CGS unit of G. After this, unit the relations between the CGs and SI units and find the value of G in SI system.

Formula used:
1cm=10−2m1cm={{10}^{-2}}m
1g=10−3kg1g={{10}^{-3}}kg.

Complete step by step answer:
Let us first understand what the gravitational constant (G) is.

The gravitational constant (G) is the universal proportionality constant used in the equation for the gravitational force between the two bodies.

Suppose there are two bodies of masses m1{{m}_{1}} , m2{{m}_{2}} and separated by a distance r is given as F=Gm1m2r2F=\dfrac{G{{m}_{1}}{{m}_{2}}}{{{r}^{2}}} ….. (i).

From the equation (i) we get that G=Fr2m1m2G=\dfrac{F{{r}^{2}}}{{{m}_{1}}{{m}_{2}}}.

Let us calculate the CGS unit of G.
The CGS unit of force (F) is gcms−2gcm{{s}^{-2}} .
The CGS unit of distance (r) is cm.
The CGS unit of mass (m) is g.
Here, the units g, cm and s are gram, centimetre and second respectively.

Therefore, the CGS unit of G is gcms−2(cm)2g2=g−1cm3s−2\dfrac{gcm{{s}^{-2}}{{(cm)}^{2}}}{{{g}^{2}}}={{g}^{-1}}c{{m}^{3}}{{s}^{- 2}}

This means that the value of G in CGS system is G=6.7×10−8g−1cm3s−2G=6.7\times {{10}^{-8}}{{g}^{-1}}c{{m}^{3}}{{s}^{- 2}} ….. (i)

Let us use the following relations between the CGS and SI units to calculate the value of G in SI system.
1cm=10−2m1cm={{10}^{-2}}m
1g=10−3kg1g={{10}^{-3}}kg.
Substitute the values in (i).
⇒G=6.7×10−8(10−3kg)−1(10−2m)3s−2\Rightarrow G=6.7\times {{10}^{-8}}{{({{10}^{-3}}kg)}^{-1}}{{({{10}^{-2}}m)}^{3}}{{s}^{-2}}
⇒G=6.7×10−11kg−1m3s−2\Rightarrow G=6.7\times {{10}^{-11}}k{{g}^{-1}}{{m}^{3}}{{s}^{-2}}.

This means that the value of the gravitational constant G in SI system is 6.7×10−11kg−1m3s−26.7\times {{10}^{- 11}}k{{g}^{-1}}{{m}^{3}}{{s}^{-2}}.

Note: We can also write the CGS unit of force as dyne, where 1dyne=1gcms−21dyne=1gcm{{s}^{-2}}.

Therefore, we can write the unit of G as dyn(cm2)(g−2)dyn(c{{m}^{2}})({{g}^{-2}}).
The SI unit is called Newton (N) and 1N=1kgms−21N=1kgm{{s}^{-2}}.

If we use the above relation between the CGS and SI units, then we will get that 1N=105dyn1N={{10}^{5}}dyn.