Explain error of a sum or a difference.
Let two physical quantities $\mathrm{A}$ and $\mathrm{B}$ has measured value $\mathrm{A} \pm \Delta \mathrm{A}$ and $\mathrm{B} \pm \Delta \mathrm{B}$ where, $(i)$ For addition :
Let $Z$ is quantity obtained by addition of $A$ and $B$.
$\therefore \mathrm{Z}=\mathrm{A}+\mathrm{B}$
Let error is $Z$ be $\Delta Z$
$Z \pm \Delta Z=(A \pm \Delta A)+(B \pm \Delta B)$
$A+B=Z$
$\therefore \pm \Delta Z=\pm \Delta A \pm \Delta B$
For maximum absolute error,
$\Delta Z=\Delta A+\Delta B$
$(ii)$ For Subtraction :
Let difference of $A$ and $B$ is $Z$
$\therefore Z=A -B[\text { Let } A>B]$
$Z \pm \Delta Z =(A \pm \Delta A)-(B \pm \Delta B)$
$=(A-B)-(\pm \Delta A \pm \Delta B)$
$A-B =Z$
$\pm \Delta Z =\pm \Delta A \pm \Delta B$
$For maximum error in $Z$,$
$\Delta Z=\Delta A+\Delta B$
The period of oscillation of a simple pendulum is given by $T = 2\pi \sqrt {\frac{l}{g}} $ where $l$ is about $100 \,cm$ and is known to have $1\,mm$ accuracy. The period is about $2\,s$. The time of $100$ oscillations is measured by a stop watch of least count $0.1\, s$. The percentage error in $g$ is ......... $\%$
A physical quantity $X$ is related to four measurable quantities $a,\, b,\, c$ and $d$ as follows $X = a^2b^3c^{\frac {5}{2}}d^{-2}$. The percentange error in the measurement of $a,\, b,\, c$ and $d$ are $1\,\%$, $2\,\%$, $3\,\%$ and $4\,\%$ respectively. What is the percentage error in quantity $X$ ? If the value of $X$ calculated on the basis of the above relation is $2.763$, to what value should you round off the result.
The radius of a sphere is $(5.3 \pm 0.1) \,cm$. The percentage error in its volume is
A student performs an experiment for determination of $g \left(=\frac{4 \pi^{2} l }{ T ^{2}}\right), \ell =1 m$ and he commits an error of $\Delta \ell$. For $T$ he takes the time of $n$ oscillations with the stop watch of least count $\Delta T$ and he commits a human error of $0.1 s$ For which of the following data, the measurement of $g$ will be most accurate?
While measuring the acceleration due to gravity by a simple pendulum, a student makes a positive error of $1\%$ in the length of the pendulum and a negative error of $3\%$ in the value of time period. His percentage error in the measurement of $g$ by the relation $g = 4{\pi ^2}\left( {l/{T^2}} \right)$ will be ........ $\%$