Three equal weights of mass m each are hanging on a string passing over a fixed pulley as shown in Fig. What are the tensions in the string connecting weights A to B and B to C

 

${\text{T}}_{\text{1}}-\text{mg}=\text{ma}\to \left(1\right)$

${\text{T}}_2+\text{mg}-{\text{T}}_{\text{1}}=\text{ma}\to \left(2\right)$

$\text{mg}-{\text{T}}_{\text{2}}=\text{ma}\to \left(3\right)$

${\text{T}}_{\text{1}}-\text{mg}=\text{ma}$

$\underset{\_}{{\text{T}}_2-{\text{T}}_{\text{1}}+\text{mg}=\text{ma}}$

${\text{T}}_{\text{2}}=2\text{ma}$

$\underset{\_}{\text{mg}-{\text{T}}_{\text{2}}=\text{ma}}$                         

$\text{mg}=3\text{ma}$

$\text{g}=\text{3a}$

$\frac{\text{g}}{\text{3}}=\text{a}$

${\text{T}}_1-\text{mg}=\text{m}\text{.}\frac{\text{g}}{\text{3}}$

${\text{T}}_{\text{1}}=\text{mg}+\frac{\text{mg}}{\text{3}}$

${\text{T}}_1=\frac{3\text{mg}+\text{mg}}{3}$

${\text{T}}_{\text{1}}=\text{4mg/3}$

${\text{T}}_1=\frac{4}{3}\text{mg}$

${\text{T}}_2=\text{mg}-\frac{\text{mg}}{\text{3}}$

${\text{T}}_2=\frac{3\text{mg}-\text{mg}}{3}$

${\text{T}}_{\text{2}}=\frac{\text{2mg}}{\text{3}}$

${\text{T}}_{\text{1}}=\frac{4}{3}\text{mg}$

${\text{T}}_2=\frac{2}{3}\text{mg}$