Question

Define Universal gravitational constant G. What is the dimensional formula G?

Answer 1

Hint: Newton’s Law of gravitation:
Newton’s Law of Universal Gravitation states that every particle attracts every other particle in the universe with a force that is directly proportional to the product of the masses and inversely proportional to the square of the distance between them.
$F \propto \dfrac{{{m_1}{m_{_2}}}}{{{r^2}}}$
To remove the proportionality we always need to introduce a constant of proportionality so here newton inserted a constant called Universal Gravitational constant $\left( G \right)$
\[F = G\dfrac{{{m_1}{m_{_2}}}}{{{r^2}}}\]

Complete solution:
Universal gravitational constant
The gravitational constant is the proportionality constant that is used in Newton’s Law of gravitation. The force of attraction between any two unit masses separated by a unit distance is called the universal gravitational constant denoted by $\left( G \right)$measured in\[\left( {\dfrac{{N{m^2}}}{{k{g^2}}}} \right)\].
Mathematically,
It two objects of unit mass and are separated by unit distance then, the force with which they’ll attract each other is called universal gravitational constant $\left( G \right)$
\[{m_1},{m_{_2}} = 1\]Unit
$r = 1$Unit
$\therefore F = G$

Dimensional formula of Universal Gravitational Constant$\left( G \right)$
The expressions or formulae which tell us how and which of the fundamental quantities are present in a physical quantity are known as the Dimensional Formula of the Physical Quantity.
Suppose there is a physical quantity X which depends on base dimensions M (Mass), L (Length), and T (Time) with respective powers a, b and c, then its dimensional formula is represented as: \[\left[ {{M^a}{L^b}{T^c}} \right]\]
Dimensional formula for basic physical quantities,
Mass = $\left[ M \right]$
Distance = \[\left[ L \right]\]
Time = $\left[ T \right]$
Velocity: It is the distance covered per unit time so $v = \left( {\dfrac{d}{t}} \right)$
From here we can say that is the ratio of distance and times so the dimensional formula will be:
$\dfrac{L}{T}$ (Or) $\left[ {L{T^{ - 1}}} \right]$
Acceleration = it is the rate of change of velocity with time so,
We can say that $a = \dfrac{v}{t}$
Here velocity is divided by time again we will substitute the dimensional formulas of velocity band time to get the dimensional formula for acceleration
\[
  \therefore a = \left[ {\dfrac{{L{T^{ - 1}}}}{T}} \right] \\
  \Rightarrow a = \left[ {L{T^{ - 2}}} \right] \\
 \]
 Force: It is defined as the mass time of acceleration so,
$F = M \times A$
Now we will be using the dimensional formula of acceleration and mass to find out the dimensional formula of force
$\because F = M \times A$
The dimensional formula of force will also be the product of the dimensional formula of the other two quantities
So, $
  F = M \times L{T^{ - 2}} \\
  \Rightarrow F = \left[ {ML{T^{ - 2}}} \right] \\
 $
 So now as we know,
\[
  F = G\dfrac{{{m_1}{m_{_2}}}}{{{r^2}}} \\
  \therefore G = \dfrac{{F{r^2}}}{{{m_1}{m_{_2}}}} \\
 \]
Now for the dimensional formula of $\left( G \right), $substitute the dimensional formula of other quantities
As r is the radius so its dimensional formula will be the same as of distance i.e. $\left( L \right)$
\[
  G = \left[ {\dfrac{{\left( {ML{T^{ - 2}}} \right)\left( {{L^2}} \right)}}{{\left( {M \times M} \right)}}} \right] \\
  \Rightarrow G = \left[ {\dfrac{{{L^3}{T^{ - 2}}}}{M}} \right] \\
  \Rightarrow G = \left[ {{M^{ - 1}}{L^3}{T^{ - 2}}} \right] \\
 \]
On solving we get,
Dimensional formula for $\left[ G \right]$=$\left[ {{M^{ - 1}}{L^3}{T^{ - 2}}} \right]$.

Final answer is, the dimensional formula for universal gravitation constant is $\left[ {{M^{ - 1}}{L^3}{T^{ - 2}}} \right]$.

Note: 1. Dimensional formula of any quantity can be derived for the fundamental quantities if the relation between them is known.
2. Dimensional Formulas are used to check whether a given formula is dimensionally correct or not.
3. Dimensional Formulae become not defined in the case of the trigonometric, logarithmic, and exponential functions as they are not physical quantities.