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.
$F\propto {\displaystyle \frac{m_1{m}_{{}_2}}{r^2}}$
To remove the proportionality we always need to introduce a constant of proportionality so here newton inserted a constant called Universal Gravitational constant $\left(G\right)$
$$F=G{\displaystyle \frac{m_1{m}_{{}_2}}{r^2}}$$

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$$\left({\displaystyle \frac{N{m}^2}{k{g}^2}}\right)$$.
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 $v=\left({\displaystyle \frac{d}{t}}\right)$
From here we can say that is the ratio of distance and times so the dimensional formula will be:
${\displaystyle \frac{L}{T}}$ (Or) $\left[L{T}^{-1}\right]$
Acceleration = it is the rate of change of velocity with time so,
We can say that $a={\displaystyle \frac{v}{t}}$
Here velocity is divided by time again we will substitute the dimensional formulas of velocity band time to get the dimensional formula for acceleration
$$\begin{gathered} \therefore a=\left[{\displaystyle \frac{L{T}^{-1}}{T}}\right] \\ \Rightarrow a=\left[L{T}^{-2}\right] \end{gathered}$$
 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, $$\begin{gathered} F=M\times L{T}^{-2} \\ \Rightarrow F=\left[ML{T}^{-2}\right] \end{gathered}$$
 So now as we know,
$$\begin{gathered} F=G{\displaystyle \frac{m_1{m}_{{}_2}}{r^2}} \\ \therefore G={\displaystyle \frac{F{r}^2}{m_1{m}_{{}_2}}} \end{gathered}$$
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. $\left(L\right)$
$$\begin{gathered} G=\left[{\displaystyle \frac{\left(ML{T}^{-2}\right)\left(L^2\right)}{\left(M\times M\right)}}\right] \\ \Rightarrow G=\left[{\displaystyle \frac{L^3{T}^{-2}}{M}}\right] \\ \Rightarrow G=\left[M^{-1}{L}^3{T}^{-2}\right] \end{gathered}$$
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.