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Question: A physical quantity \(X\) is represented by \(X=\left[ {{M}^{\eta }}{{L}^{-\theta }}{{T}^{-\phi }} \...

A physical quantity XX is represented by X=[MηL−θT−ϕ]X=\left[ {{M}^{\eta }}{{L}^{-\theta }}{{T}^{-\phi }} \right] . The maximum percentage errors in the measurement of MM ,LL and TT respectively are α\alpha %,\beta % andγ\gamma % . The maximum percentage error in the measurement of XX will be.
A. (ηα−θβ−γα)\left( \eta \alpha -\theta \beta -\gamma \alpha \right)%
B. (θβ+ϕγ−ηα)\left( \theta \beta +\phi \gamma -\eta \alpha \right)%
C. (αη−βθ−γϕ)\left( \dfrac{\alpha }{\eta }-\dfrac{\beta }{\theta }-\dfrac{\gamma }{\phi } \right)%
D. (ηα+θβ+ϕγ)\left( \eta \alpha +\theta \beta +\phi \gamma \right)%

Explanation

Solution

Hint Using the maximum expected percentage error formula.
∣δXX∣max⁡×100=(ηδMM×100)+(θδLL×100)+(ϕδTT×100){{\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(ρ)\left( \rho \right) of a metal block ,we have to measure mass(m)\left( m \right) and its volume (v)\left( v \right) which are related to ρ\rho by a formula ρ=mv\rho =\dfrac{m}{v} .The accuracy in the value of ρ\rho depends upon the accuracy of measurement of mm and vv .
Given,
ΔMM×100=α\dfrac{\Delta M}{M}\times 100=\alpha %
ΔLL×100=β\dfrac{\Delta L}{L}\times 100=\beta %
ΔTT×100=γ\dfrac{\Delta T}{T}\times 100=\gamma %
For maximum percentage error ,
∣ΔXX∣max⁡×100=η(ΔMM×100)+θ(ΔLL×100)+ϕ(ΔTT×100){{\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)
∣δXX∣max⁡×100=(ηα+θβ+γα){{\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.