Question

Dimensions of an unknown quantity $\varphi =\frac{ma}{\alpha }log\left(1+\frac{\alpha l}{ma}\right)$ where m = mass, a = acceleration and I = length are

• A. $\left[ML{T}^{-2}\right]$
• B. $\left[M{T}^{-2}\right]$
• C. $\left[M^{0}L{T}^0\right]$
• D. $\left[M{L}^{-3}\right]$

Answer 1

Answer: (C)

Logarithm has no dimensions

$\therefore 1+\frac{\alpha l}{ma}=$ dimensionless

$\therefore \frac{\alpha l}{ma}=\left[M^0{L}^0{T}^0\right]$

$\Rightarrow \alpha =\frac{ma}{l}=\frac{\left[ML{T}^{-2}\right]}{\left[L\right]}=\left[M{T}^{-2}\right]$

$\because$ Dimension of $\varphi =$ Dimensions of $\frac{ma}{\alpha }$

$=\frac{\left[ML{T}^{-2}\right]}{M{T}^{-2}}=\left[L\right]=\left[M^{0}L{T}^0\right]$