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 $X, \frac{\Delta X }{ X } \times 100$
$= a \alpha+ b \beta+ c \gamma$