The dimensions of universal qravitational constant are

The correct answer is Option 1: [M⁻¹L³T⁻²].

To find the dimensions of the universal gravitational constant ($G$), we start with Newton's law of universal gravitation. The gravitational force ($F$) between two masses ($m_1$ and $m_2$) separated by a distance ($r$) is given by the formula:
$F=\frac{G{m}_1{m}_2}{r^2}$

Rearranging this formula to solve for the gravitational constant $G$, we get:
$G=\frac{F{r}^2}{m_1{m}_2}$

Now, we substitute the dimensional formulas of the individual physical quantities into this expression:

• Force ($F$) has dimensions: $\left[{\mathrm{MLT}}^{-2}\right]$
• Distance ($r$) has dimensions: $\left[L\right]$, so $r^2$ has dimensions: $\left[L^2\right]$
• Masses ($m_1$ and $m_2$) have dimensions: $\left[M\right]$, so their product $m_1{m}_2$ has dimensions: $\left[M^2\right]$

Substituting these dimensions into the rearranged equation:
$\left[G\right]=\frac{\left[{\mathrm{MLT}}^{-2}\right]\left[L^2\right]}{\left[M^2\right]}$

Combining the terms in the numerator:
$\left[G\right]=\frac{\left[{\mathrm{ML}}^3{T}^{-2}\right]}{\left[M^2\right]}$

Simplifying the powers of $M$ by bringing the denominator to the numerator, we obtain:
$\left[G\right]=\left[M^{1-2}{L}^3{T}^{-2}\right]$
$\left[G\right]=\left[M^{-1}{L}^3{T}^{-2}\right]$

Thus, the dimensional formula of the universal gravitational constant is [M⁻¹L³T⁻²].