In the measurement of physical quantity X=(A^2 B)/(C^1 / 3 D), the percentage error introduced i

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In the measurement of physical quantity X=(A^2 B)/(C^1 / 3...

In the measurement of physical quantity $X=\frac{A^{2} B}{C^{1 / 3} D}$, the percentage error introduced in the measurements of the quantities $A, B, C$ and $D$ are $2 \%, 2 \%, 4 \%$ and $5 \%$, respectively. Then the minimum amount of percentage error in the measurement of $X$ is contributed by (a) $A$ (b) $\underline{B}$ (c) $C$ (d) $D$

Key Concepts

Logarithmic Differentiation in Error Analysis

Logarithmic differentiation is a method used in error analysis to simplify the calculation of percentage errors in functions where variables are multiplied, divided, or raised to a power. By taking the logarithm of the quantity and differentiating, the relative errors can be easily represented as the sum or difference of the individual relative errors multiplied by their respective exponents.

Error Propagation

This concept deals with determining how uncertainties in measured quantities affect the overall uncertainty in a derived quantity. When a physical quantity depends on several measured variables, the errors in those variables combine in a way that depends on their mathematical relationship, typically by adding relative uncertainties in multiplicative or power law functions.

Relative (Percentage) Error

Relative or percentage error represents the uncertainty of a measurement expressed as a percentage of the measured value. It is used to compare the significance of uncertainty between different quantities, regardless of their magnitudes, which is crucial when combining errors in a formula.

Effect of Exponents on Error Propagation

When a measured quantity is raised to a power, its percentage error is multiplied by the absolute value of that exponent. This means that even if the measurement error is small, applying an exponent can amplify or reduce its impact on the overall error in a derived quantity.

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