Smaller mass, $m_{1}=8 \,kg$
Larger mass, $m_{2}=12\, kg$
Tension in the string $=T$
Mass $m_{2},$ owing to its weight, moves downward with acceleration $a,$ and mass $m_{1}$ moves upward.
Applying Newton's second law of motion to the system of each mass:
For mass $m_{\underline{1}}:$ The equation of motion can be written as:
$T-m_{1} g =m a$
For mass $m_{2}$ : The equation of motion can be written as:
$m_{2} g -T=m_{2} a$
Adding above equations , we get:
$\left(m_{2}-m_{1}\right) g =\left(m_{1}+m_{2}\right) a$
$\therefore a=\left(\frac{m_{2}-m_{1}}{m_{1}+m_{2}}\right) g$
$=\left(\frac{12-8}{12+8}\right) \times 10=\frac{4}{20} \times 10=2 m / s ^{2}$
Therefore, the acceleration of the masses is $2 \;m / s ^{2}$
Substituting the value of $a$ in equation ( $i i$ ), we get:
$m_{2} g -T=m_{2}\left(\frac{m_{2}-m_{1}}{m_{1}+m_{2}}\right) g$
$T=\left(m_{2}-\frac{m_{2}^{2}-m_{1} m_{2}}{m_{1}+m_{2}}\right) g$
$=\left(\frac{2 m_{1} m_{2}}{m_{1}+m_{2}}\right) g$
$=\left(\frac{2 \times 12 \times 8}{12+8}\right) \times 10$
$=\frac{2 \times 12 \times 8}{20} \times 10=96\, N$
Therefore, the tension in the string is $96\, N$.
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$Reason$ : The accuracy and precision of measuring instruments along with errors in measurements should be taken into account, while expressing the result.
