Question

What are the dimensions of $a$ and $b$ in the relation $F=a\sqrt{x}+b{t}^2$ where $F$ is force, $x$ is distance and $t$ is time?

• A. $\left[M{L}^{\frac{1}{2}}{T}^{-2}\right],\left[ML{T}^{-4}\right]$
• B. $\left[M{L}^{\frac{3}{2}}{T}^{-2}\right],\left[ML{T}^{-2}\right]$
• C. $\left[M^2{L}^{\frac{1}{2}}{T}^{\frac{-1}{2}}\right],\left[M^3{L}^{\frac{1}{2}}{T}^{-2}\right]$
• D. $\left[M^3{L}^{\frac{1}{2}}{T}^{-2}\right],\left[M{L}^{\frac{3}{2}}{T}^{-2}\right]$

Answer 1

The correct option is A $\left[M{L}^{\frac{1}{2}}{T}^{-2}\right],\left[ML{T}^{-4}\right]$

Applying principle of homogeneity, every term in an algebraic equation of physical quantities must have the same unit.

Thus, each term in the equation must have the dimensions of force.

$\left[ML{T}^{-2}\right]=\left[a\right]\left[L^{1/2}\right]=\left[b\right]\left[T^2\right]$

$⟹\left[a\right]=\left[M{L}^{1/2}{T}^{-2}\right]$
$\left[b\right]=\left[ML{T}^{-4}\right]$