Question

A physical quantity is given by $X=[M^a{L}^b{T}^c]$. The percentage error in measurement of $M,L$ and $T$ are $\alpha ,\beta$ and $\gamma$ respectively. Then, the maximum % error in the quantity $X$ is :

• A. $a\alpha +b\beta +c\gamma$
• B. $a\alpha +b\beta -c\gamma$
• C. $\frac{a}{\alpha }+\frac{b}{\beta }+\frac{c}{\gamma }$
• D. none of these

Answer 1

Answer: (A)

The quantity is given as $X=\left[M^a{L}^b{T}^c\right]$
Taking logrithm on both sides $\ln X=a\ln M+b\ln L+c\ln T$
Differentiating and multiplying by $100$, we get
$\frac{\Delta X}{X}\times 100=a\frac{\Delta M}{M}\times 100+b\frac{\Delta L}{L}\times 100+c\frac{\Delta T}{T}\times 100$ ...(1)
Given: $\frac{\Delta M}{M}\times 100=\alpha$
$\frac{\Delta L}{L}\times 100=\beta$
$\frac{\Delta T}{T}\times 100=\gamma$
Thus error in $X,\frac{\Delta X}{X}\times 100$
$=a\alpha +b\beta +c\gamma$