Question

A physical quantity $X$ is represented by $X=\left[M^{\eta }{L}^{-\theta }{T}^{-\varphi }\right]$. 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. $(\eta \alpha -\theta \beta -\varphi \gamma )$%
• B. $(\theta \beta +\varphi \gamma -\eta \alpha )$%
• C. $(\frac{\alpha }{\eta }-\frac{\beta }{\theta }-\frac{\gamma }{\varphi })$%
• D. $(\eta \alpha +\theta \beta +\varphi \gamma )$%

Answer 1

The correct option is D $(\eta \alpha +\theta \beta +\varphi \gamma )$%

Given, $X=M^{\eta }{L}^{-\theta }{T}^{-\varphi }$

Differentiating, $\Delta X=\left(\eta M^{\eta -1}\Delta M\right)L^{-\theta }{T}^{-\varphi }-\left(\theta L^{-\theta -1}\Delta L\right)M^{\eta }{T}^{-\varphi }-\left(\varphi T^{-\varphi -1}\Delta T\right)M^{\eta }{L}^{-\varphi }$

So, $\frac{\Delta X}{X}=\eta \frac{\Delta M}{M}-\theta \frac{\Delta L}{L}-\varphi \frac{\Delta T}{T}$

As the error should be both positive and negative , so we can take mod

The % error in X $=|\frac{\Delta X}{X}|\times 100$

$=[\eta \frac{\Delta M}{M}+\theta \frac{\Delta L}{L}+\varphi \frac{\Delta T}{T}]\times 100$

$=(\eta \alpha +\theta \beta +\varphi \gamma )$ %