Question

Consider an equation represented by,
$b=\frac{ma}{k}\left[\sqrt{1+\frac{2kl}{ma}}\right]$, where, $m$ represents mass, $a$ represents acceleration, $l$ represents length and $k$ is a constant. The SI unit of $b$ should be

• A. $\text{s}$
• B. $\text{kg}$
• C. $\text{m}$
• D. $\text{K}$

Answer 1

The correct option is C $\text{m}$

Here, $\sqrt{1+\frac{2kl}{ma}}$ is a number, thus, it is a dimensionless quantity.
$\Rightarrow \left[\frac{2kl}{ma}\right]=\left[M^o{L}^o{T}^o\right]$
$\Rightarrow \left[k\right]=\frac{\left[m\right]\left[a\right]}{\left[l\right]}$
$\Rightarrow \left[k\right]=\left[\frac{ML{T}^{-2}}{L}\right]=\left[M{T}^{-2}\right]$
Here, the dimension of $b$ will be,
$\left[b\right]=\left[\frac{ma}{k}\right]$
$\Rightarrow \left[b\right]=\left[\frac{ML{T}^{-2}}{M{T}^{-2}}\right]=\left[L\right]$
Thus, the SI unit of $b$ will be metre.

Why this question? It tests your understanding of the principle of homogeneity in dimensional analysis.