Question 224: Units , Dimensions and Error Analysis

Question

A physical quantity C is related to four other quantities $\mathrm{p}, \mathrm{q}, \mathrm{r}$ and s as follows
$$ C=\frac{p q^2}{r^3 \sqrt{s}} $$
The percentage errors in the measurement of $\mathrm{p}, \mathrm{q}, \mathrm{r}$ and s are $1 \%, 2 \%, 3 \%$ and $2 \%$, respectively. The percentage error in the measurement of $C$ will be__________%

Accepted Answer

To determine the percentage error in the measurement of $ C $, which is related to $ p $, $ q $, $ r $, and $ s $ as:
$  C = \frac{p q^2}{r^3 \sqrt{s}}  $
we first express it in terms of powers:
$  C = p^1 q^2 r^{-3} s^{-1/2}  $
The percentage error in $ C $ can be calculated using the formula for the propagation of error, which is:
$  \left(\frac{\Delta C}{C}ight)_{\max} = \left|\frac{\Delta p}{p}ight| + 2\left|\frac{\Delta q}{q}ight| + 3\left|\frac{\Delta r}{r}ight| + \frac{1}{2}\left|\frac{\Delta s}{s}ight| $
Given the percentage errors for $ p $, $ q $, $ r $, and $ s $ are $ 1\% $, $ 2\% $, $ 3\% $, and $ 2\% $ respectively, we substitute these values into the formula:
$  \left(\frac{\Delta C}{C}ight)_{\max} = 1\% + 2 	imes 2\% + 3 	imes 3\% + \frac{1}{2} 	imes 2\% $
Calculating each part:
Contribution from $ p $: $ 1\% $
Contribution from $ q $: $ 4\% $ (since $ 2 	imes 2\% = 4\% $)
Contribution from $ r $: $ 9\% $ (since $ 3 	imes 3\% = 9\% $)
Contribution from $ s $: $ 1\% $ (since $ \frac{1}{2} 	imes 2\% = 1\% $)
Adding these contributions together:
$  1\% + 4\% + 9\% + 1\% = 15\% $
Thus, the maximum percentage error in the measurement of $ C $ is $ 15\% $.

Units , Dimensions and Error Analysis

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