combination of errors

Question

The period of oscillation of a simple pendulum is T = 2π√(L/g). If the error in measurement of L is 2% and that in g is 3%, what is the percentage error in T?

RankGuru · Accepted Answer

1. **Error Rule for Powers**: For $Z = X^n / Y^m$, the absolute relative error sum is $\frac{\Delta Z}{Z} = n \frac{\Delta X}{X} + m \frac{\Delta Y}{Y}$.
2. **Identify Powers**: $T = 2\pi L^{1/2} g^{-1/2} \implies n=1/2, m=1/2$.
3. **Calculation**:
   - $\% 	ext{ error in } T = \frac{1}{2} (\% 	ext{ error in } L) + \frac{1}{2} (\% 	ext{ error in } g)$.
   - $\% 	ext{ error in } T = 0.5(2\%) + 0.5(3\%) = 1\% + 1.5\%$.
4. **Result**: Total percentage error is **2.5%**.

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combination of errors

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