Explain uncertainty or error in given measurement by suitable example.

$(1)$ Length and breadth of a thin rectangular plate is $l=16.2 \mathrm{~cm}$ and breadth $b=10.1 \mathrm{~cm}$ Least count of meterscale is $0.1 \mathrm{~cm}$ hence absolute error in measurement should be $0.1 \mathrm{~cm}$.

$l=(16.2 \pm 0.1) \mathrm{cm}$

$b=(10.1 \pm 0.1) \mathrm{cm}$

$\%$ error in measurement of length,

$l=(16.2 \pm 0.6 \%) \mathrm{cm}$

$\%$ error in measurement of breadth,

$b=(10.1 \pm 1 \%) \mathrm{cm}$

Area of rectangular plate,

$\mathrm{A} =l b$

$=16.2 \times 10.1=163.62 \mathrm{~cm}^{2}$

$\%$ error in $\mathrm{A}$ :

$=\frac{\Delta \mathrm{A}}{\mathrm{A}} \times 100=\frac{\Delta l}{l} \times 100+\frac{\Delta b}{b} \times 100$

$=(0.6 \%+1 \%)=1.6 \%$

$\Delta \mathrm{A} =\frac{1.6 \times 163.62}{100}=2.6$

Area : $\mathrm{A}=(163.62 \pm 2.6) \mathrm{cm}^{2}$

Here, minimum significant digit are $3$ hence area should be represented as, $\mathrm{A} \approx 163 \pm 3 \mathrm{~cm}^{2}$

$\therefore$ Error in measurement of area of plate is $3 \mathrm{~cm}^{2}$.

$(2)$ If a set of experimental data is specified to ' $n$ ' significant figures, a result obtained by combining the data will also be valid to ' $n$ ' significant figures.

However, if data is subtracted the number of significant figures can be reduced. For example,

$12.9 \mathrm{~g}-7.06 \mathrm{~g}=5.84 \mathrm{~g}$

Here, there are $3$ significant digits. But in subtraction digit after decimal point are considered hence this will be represented as $5.8 \mathrm{~g}$.