Question

A physical quantity $X$ is represented by $X=\left[ {{M}^{\eta }}{{L}^{-\theta }}{{T}^{-\phi }} \right]$ . The maximum percentage errors in the measurement of $M$ ,$L$ and $T$ respectively are $\alpha %,\beta %$ and$\gamma %$ . The maximum percentage error in the measurement of $X$ will be.
A. $\left( \eta \alpha -\theta \beta -\gamma \alpha \right)%$
B. $\left( \theta \beta +\phi \gamma -\eta \alpha \right)%$
C. $\left( \dfrac{\alpha }{\eta }-\dfrac{\beta }{\theta }-\dfrac{\gamma }{\phi } \right)%$
D. $\left( \eta \alpha +\theta \beta +\phi \gamma \right)%$

Answer 1

HintUsing the maximum expected percentage error formula.
 ${{\left| \dfrac{\delta X}{X} \right|}_{\max }}\times 100=\left( \eta \dfrac{\delta M}{M}\times 100 \right)+\left( \theta \dfrac{\delta L}{L}\times 100 \right)+\left( \phi \dfrac{\delta T}{T}\times 100 \right)$

 Complete step-by-step solution:Error means that uncertainty of measurement is technically called an “error”. Error is always expressed in percentage.
To determine a physical quantity we have to measure various quantities. Which are related to that physical quantity by a formula? For example, to determine the density$\left( \rho \right)$ of a metal block ,we have to measure mass$\left( m \right)$ and its volume $\left( v \right)$ which are related to $\rho $ by a formula $\rho =\dfrac{m}{v}$ .The accuracy in the value of $\rho $ depends upon the accuracy of measurement of $m$ and $v$ .
Given,
$\dfrac{\Delta M}{M}\times 100=\alpha %$
$\dfrac{\Delta L}{L}\times 100=\beta %$
$\dfrac{\Delta T}{T}\times 100=\gamma %$
For maximum percentage error ,
${{\left| \dfrac{\Delta X}{X} \right|}_{\max }}\times 100=\eta \left( \dfrac{\Delta M}{M}\times 100 \right)+\theta \left( \dfrac{\Delta L}{L}\times 100 \right)+\phi \left( \dfrac{\Delta T}{T}\times 100 \right)$
${{\left| \dfrac{\delta X}{X} \right|}_{\max }}\times 100=\left( \eta \alpha +\theta \beta +\gamma \alpha \right)%$

Note:When we calculate the maximum error then we add all the terms without sign but sometimes we write these terms with sign , if it has negative sign ,then it is wrong.