A pendulum of length

PHYSICS · Subjective · Fermi University Physics · Scientific Practice & Experimental Methods · Mathematical Methods for Physics · Uncertainty in measurement

Computational Physics, Ch Numbers a machine can hold, and the two errors that follow

Problem

L = 2.50\,\mathrm{m}

, known to a relative uncertainty of

0.40\%

, has its period estimated from

T = 2\pi\sqrt{L/g}

with

g = 9.81\,\mathrm{m\,s^{-2}}

taken as exact. A program prints the result as

T = 3.171869924610\,\mathrm{s}

, and its author claims that printing all those digits makes the answer more precise. (a) Say what is wrong with the claim, in terms of where the sixteen digits of a stored binary64 value come from and what they can and cannot know about

L

. (b) Compute the condition number of

T

as a function of

L

, use it to bound the relative uncertainty in

T

, and quote

T

to the number of digits this supports.

Browse the problem library