Question

Discuss the process of combination of errors when two physical quantities are multiplied.

Solution

Combination error when two quantities are multiplied, is given by
Let X = AB
Let ∆A, ∆B and ∆X be the absolute errors in A, B and X respectively.
Therefore the above equation with error can be written as,
$space space space space space space space space space space space space space space space space straight X plus-or-minus straight capital delta equals left parenthesis straight A plus-or-minus ΔA right parenthesis left parenthesis straight B plus-or-minus ΔB right parenthesis space space space space space rightwards double arrow space space space space straight X open parentheses 1 plus-or-minus ΔX over straight X close parentheses equals AB open parentheses 1 plus-or-minus ΔA over straight A close parentheses open parentheses 1 plus-or-minus ΔB over straight B close parentheses space space space space space$

$space space space space space space space space space space space space rightwards double arrow space space space space space space straight X open parentheses 1 plus-or-minus ΔX over straight X close parentheses equals AB open parentheses 1 plus-or-minus ΔA over straight A plus-or-minus ΔB over straight B plus-or-minus ΔA over straight A ΔB over straight B close parentheses rightwards double arrow space space space space space open parentheses 1 plus-or-minus ΔX over straight X close parentheses equals open parentheses 1 plus-or-minus ΔX over straight X plus-or-minus ΔB over straight B plus-or-minus ΔA over straight A ΔB over straight B close parentheses$
$<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$ are small quantities.

Therefore the product $ΔA over straight A. ΔB over straight B$ will be very small and hence can be neglected.
Thus the above equation reduce to,
$space space space space space space space plus-or-minus ΔX over straight X equals plus-or-minus ΔA over straight A plus-or-minus ΔB over straight B$
Maximum value of fractional error in X is
$ΔX over straight X equals ΔA over straight A plus ΔB over straight B$
Multiplying both sides by 100
$<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$
i.e. Maximum possible %age error in X = Maximum possibe %age error in A + Maximum possible %age error in B.

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Units And Measurement

Discuss the process of combination of errors when two physical quantities are multiplied.

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