In the following figures (i) and (ii), all the surfaces are smooth string is massless and pulley is ideal. The mass of the rope is m in both the cases. If $T_{1}$ and $T_{2}$ be the tensions at the mid points of the ropes in figure (i) and figure (ii) respectively, then $T_{1} : T_{2}$ is

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T = ma Mg − T = Ma ∴ Mg = ( M + m ) a ∴ a = ( M + m ) Mg ​ ∴ T 1 ​ = 2 m ​ × ( M + m ) M ​ × g Now a 2 ​ = m M ​ g ∴ T 2 ​ = 2 m ​ × m M ​ g ∴ T 1 ​ : T 2 ​ = ( M + m ) 1 ​ : m 1 ​ = m : ( M + m )

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