maximum error percentage

**Dr Zoidburg** · September 4th 2008, 10:40 PM

period T of a simple pendulum with small oscillations is calculated:

$T=2\pi \sqrt{\frac {L}{g}}$

L = length of pendulum, with 0.5% max error value
g = acceleration due to gravity, with 0.1% max error value

Use differentiation to approximate the max % error in the calculated value of T.

any help gratefully received!

**flyingsquirrel** · September 4th 2008, 11:50 PM

Hello,

Originally Posted by Dr Zoidburg

period T of a simple pendulum with small oscillations is calculated:

$T=2\pi \sqrt{\frac {L}{g}}$

L = length of pendulum, with 0.5% max error value
g = acceleration due to gravity, with 0.1% max error value

Use differentiation to approximate the max % error in the calculated value of T.

any help gratefully received!

We have $T=2\pi \sqrt{\tfrac {L}{g}}$ so

$\ln T=\ln\left( 2\pi\right) + \ln \sqrt{L}-\ln \sqrt{g}=\ln\left( 2\pi\right) + \frac{\ln L}{2}-\frac{\ln g}{2}$ .

Differentiate :

$\frac{\mathrm{d}T}{T} = 0 + \frac12\cdot\frac{\mathrm{d}L}{L} - \frac12\cdot\frac{\mathrm{d}g}{g}=\frac12\cdot\fra c{\mathrm{d}L}{L} - \frac12\cdot\frac{\mathrm{d}g}{g}$

Can you take it from here ?

**Dr Zoidburg** · September 5th 2008, 03:54 AM

I think so.
Is
$\frac{dL}{L} = \frac{1}{2L}$
and
$\frac{dg}{g} = -\frac{1}{2g}$
?

**flyingsquirrel** · September 5th 2008, 04:22 AM

Originally Posted by Dr Zoidburg

I think so.
Is
$\frac{dL}{L} = \frac{1}{2L}$
and
$\frac{dg}{g} = -\frac{1}{2g}$
?

No, you're given relative errors, not absolute ones. From

$\frac{\mathrm{d}T}{T}=\frac12\cdot\frac{\mathrm{d} L}{L} - \frac12\cdot\frac{\mathrm{d}g}{g}$

we derive

$\frac{\Delta T}{T}=\frac12\cdot\frac{\Delta L}{L} {\color{blue}+} \frac12\cdot\frac{\Delta g}{g}$

where $\Delta T$ , $\Delta L$ , $\Delta g$ are the absolute error in $T$ , $L$ and $g$ , respectively. By definition, the ratios $\tfrac{\Delta T}{T}$ , $\tfrac{\Delta L}{L}$ and $\tfrac{\Delta g}{g}$ are relative errors ; as we are given $\tfrac{\Delta L}{L}=0.5\%$ and $\tfrac{\Delta g}{g}=0.1\%$ , we can compute the value of $\tfrac{\Delta T}{T}$ .

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