In an experiment to determine the Young's modulus Y of a metal wire using the formula Y = … — JAMB UTME

In an experiment to determine the Young's modulus $Y$ of a metal wire using the formula $Y = \frac{4FL}{\pi d^2 e}$ , the force $F$ , length $L$ , diameter $d$ , and extension $e$ are measured with maximum percentage errors of $1.2\%$ , $0.8\%$ , $1.5\%$ , and $2.0\%$ respectively. What is the maximum percentage error in the calculated value of $Y$ ?

Cevap: 7 %

Cevap

The maximum percentage error in the calculated value of Young's modulus

Y

7.0\%

The maximum percentage error is determined by adding the percentage error of each measured quantity, scaled by the absolute value of its power. For

Y = \frac{4FL}{\pi d^2 e}

, the calculation is

1.2\% + 0.8\% + 2(1.5\%) + 2.0\% = 7.0\%

Adım Adım Çözüm

Set up the relative error propagation formula for the derived quantity

Y

\frac{\Delta Y}{Y} = \frac{\Delta F}{F} + \frac{\Delta L}{L} + 2\left(\frac{\Delta d}{d}\right) + \frac{\Delta e}{e}

Mathematical constants (

4

and

\pi

) carry no measurement error, and exponents multiply the fractional error of their respective variables.

Convert fractional errors into percentage errors by multiplying each term by

100\%

\text{Percentage Error}(Y) = \text{Percentage Error}(F) + \text{Percentage Error}(L) + 2 \times \text{Percentage Error}(d) + \text{Percentage Error}(e)

Percentage error is directly proportional to fractional error.

Substitute the given percentage errors into the formula and sum them up.

\text{Percentage Error}(Y) = 1.2\% + 0.8\% + 2(1.5\%) + 2.0\% = 7.0\%

To find the worst-case (maximum) error, all individual percentage contributions are added regardless of whether the variable appears in the numerator or denominator.

Anahtar Kavram

Error Propagation in Derived Physical Quantities

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