Question

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

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

Answer 1

Answer: (C)

Logarithm has no dimensions
$\therefore 1+\frac{\alpha l}{ma}=$ dimensionless
$\therefore \frac{\alpha l}{ma}=\left[M^{0}L^{0}T^{0}\right]$
$\Rightarrow \alpha=\frac{ma}{l}=\frac{\left[MLT^{-2}\right]}{\left[L\right]}=\left[MT^{-2}\right]$
$\because$ Dimension of $\phi=$ Dimensions of $\frac{ma}{\alpha}$
$=\frac{\left[MLT^{-2}\right]}{MT^{-2}}=\left[L\right]=\left[M^{0}LT^{0}\right]$