A quantity $Q$ is formulated as $X^{-2} Y^{+3/2} Z^{-2/5}$. $X, Y$, 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 ________ .
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Solution Steps
1
Write the General Error Formula
Given the relation: $Q = X^{-2} Y^{3/2} Z^{-2/5}$.
The maximum fractional error in $Q$ is given by the sum of fractional errors of each component multiplied by the absolute value of their respective powers.
2
Derive the Fractional Error Expression
$$\frac{\Delta Q}{Q} = \left| -2 \right| \frac{\Delta X}{X} + \left| \frac{3}{2} \right| \frac{\Delta Y}{Y} + \left| -\frac{2}{5} \right| \frac{\Delta Z}{Z}$$
$$\frac{\Delta Q}{Q} = 2 \frac{\Delta X}{X} + \frac{3}{2} \frac{\Delta Y}{Y} + \frac{2}{5} \frac{\Delta Z}{Z}$$
3
Substitute the Given Values
We are given the fractional errors: $\frac{\Delta X}{X} = 0.1$, $\frac{\Delta Y}{Y} = 0.2$, and $\frac{\Delta Z}{Z} = 0.5$.
4
Perform the Calculation
$$\frac{\Delta Q}{Q} = 2(0.1) + \frac{3}{2}(0.2) + \frac{2}{5}(0.5)$$
$$\frac{\Delta Q}{Q} = 0.2 + 0.3 + 0.2$$
5
Find the Result
Adding the values together: $0.2 + 0.3 + 0.2 = 0.7$.
Final Answer: 0.7
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Theory
1. Fractional and Percentage Errors
Fractional error is a way of expressing uncertainty in a measurement relative to the magnitude of the measurement itself. It is defined as $\Delta A / A$. When multiplied by 100, it is called the percentage error. These values are crucial in physics experiments because they provide context; an error of 1 cm is massive for a 2 cm object but negligible for a 1 km distance. In JEE problems, fractional errors are often provided as decimal values (e.g., 0.1) to simplify the propagation calculations.
2. Propagation of Errors in Products and Quotients
When a physical quantity is derived from the product or quotient of other measured quantities, the maximum fractional error is the sum of the fractional errors of the individual quantities. For a relation like $Z = AB$, the fractional error is $\Delta Z/Z = \Delta A/A + \Delta B/B$. This occurs because uncertainties add up to create the widest possible range of error. It is important to note that even for division ($Z = A/B$), we add the errors rather than subtracting them to find the “maximum” possible error.
3. Error in Quantities Raised to a Power
If a physical quantity $Q$ depends on a parameter $X$ raised to a power $n$, the fractional error in $Q$ is $n$ times the fractional error in $X$. For $Q = X^n$, $\Delta Q/Q = |n| (\Delta X/X)$. The absolute value sign is critical because error represents uncertainty, which always increases as more measurements are combined. Negative exponents (like $X^{-2}$ in the problem) contribute positive terms to the total error calculation. This principle ensures that we account for the worst-case scenario in experimental results.
4. Independent Parameters and Systematic Error
The assumption of “independent parameters” allows us to treat the source of error for each variable as unconnected. In such cases, we use a simple linear summation of individual error contributions. If the parameters were dependent, more complex statistical methods like root-sum-square (RSS) would be required. In the context of JEE, simple summation is the standard approach for “maximum error” unless specified otherwise. This methodology helps engineers and scientists predict the reliability of a final calculated value based on the precision of their tools.
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FAQs
1
Why did the negative powers become positive?
Because we are calculating the “maximum” fractional error. Uncertainty always adds up, so we take the absolute value of exponents.
2
What if I need the percentage error instead?
Simply multiply the fractional error by 100. In this case, $0.7 \times 100 = 70\%$.
3
Does the order of X, Y, Z matter?
No, since the fractional errors are summed, the order of independent parameters does not affect the final result.
4
Is there a difference between absolute error and fractional error?
Yes. Absolute error ($\Delta X$) has units; fractional error ($\Delta X/X$) is dimensionless.
5
What if the error values were negative?
Error magnitudes are always treated as positive uncertainties in these types of problems.
6
Can fractional error be greater than 1?
Yes, though it implies that the uncertainty is larger than the measurement itself, which usually means the measurement is unreliable.
7
How do I handle addition/subtraction errors?
For addition or subtraction ($Z = A \pm B$), we add the absolute errors: $\Delta Z = \Delta A + \Delta B$.
8
Why is the power 3/2 treated as 1.5?
Yes, $3/2$ is numerically $1.5$. Multiplying $1.5$ by the error $0.2$ gives $0.3$.
9
What is the most significant source of error here?
Parameters X and Z contribute $0.2$ each, while Y contributes $0.3$. Thus, Y has the highest impact on total error.
10
Is this formula valid for small errors only?
The linear propagation formula is technically an approximation derived using calculus, intended for small errors ($\Delta X \ll X$).
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