A quantity $Q$ is formulated as $X^{-2} Y^{\frac{3}{2}} Z^{\frac{-2}{5}}$. $X, Y$ and $Z$ are independent parameters which have fractional errors of $0.1, 0.2$ and $0.5$, respectively in measurement. The maximum fractional error of $Q$ is
Calculating Maximum Fractional Error for Quantity Q
The problem involves finding the maximum fractional error in a quantity $Q$ defined as $Q = X^{-2} Y^{\frac{3}{2}} Z^{\frac{-2}{5}}$. We are given the fractional errors for the independent parameters $X, Y,$ and $Z$.
Error Propagation Formula
For a quantity $Q$ expressed in terms of independent parameters $X, Y, Z$ as $Q = X^a Y^b Z^c$, the maximum fractional error $\frac{\Delta Q}{|Q|}$ is calculated using the formula:
$ \frac{\Delta Q}{|Q|} = |a| \frac{\Delta X}{|X|} + |b| \frac{\Delta Y}{|Y|} + |c| \frac{\Delta Z}{|Z|} $
Here, $\frac{\Delta X}{|X|}$, $\frac{\Delta Y}{|Y|}$, and $\frac{\Delta Z}{|Z|}$ represent the fractional errors in $X, Y,$ and $Z$, respectively.
Applying the Formula to Q
Identify the exponents and fractional errors from the problem:
- Exponent of $X$ is $a = -2$. Fractional error in $X$ is $\frac{\Delta X}{|X|} = 0.1$.
- Exponent of $Y$ is $b = \frac{3}{2}$. Fractional error in $Y$ is $\frac{\Delta Y}{|Y|} = 0.2$.
- Exponent of $Z$ is $c = \frac{-2}{5}$. Fractional error in $Z$ is $\frac{\Delta Z}{|Z|} = 0.5$.
Step-by-Step Calculation
Substitute the values into the error propagation formula:
- Calculate the contribution from $X$: $|-2| \times 0.1 = 2 \times 0.1 = 0.2$.
- Calculate the contribution from $Y$: $|\frac{3}{2}| \times 0.2 = \frac{3}{2} \times 0.2 = 1.5 \times 0.2 = 0.3$.
- Calculate the contribution from $Z$: $|\frac{-2}{5}| \times 0.5 = \frac{2}{5} \times 0.5 = 0.4 \times 0.5 = 0.2$.
- Sum the contributions to find the maximum fractional error in $Q$: $0.2 + 0.3 + 0.2 = 0.7$.
The maximum fractional error for the quantity $Q$ is $0.7$.
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Important Questions from General Physics
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