The system shown in fig, is released from rest. Calculate the tension in the string and the force exerted by the string on the pulleys, assuming pulleys and strings are massless.

Explanation

By Newton's second law, we have
`T_(1)-1g=1a` ..(i)
`T_(2)-T_(1)=3a` ..(ii)
and `2g-T_(2)=2a` ..(iii)
Solving above equation, we get
`a=g/6 ms^(-2), T_(1)=(7g)/(6)N`, and `T_(2)=(5g)/(3)N`
Force on pulley `P_(1),F_(1)=sqrt(T_(1)^(2)+T_(1)^(2))=sqrt(2(T_(1))`
Force on pulley `(P_2),(F_2)=sqrt(T_(2)^(2)+T_(2)^(2))=sqrt(2(T_(2))`
By using shortcut method:
`a=("Unbalanced load")/("Total mass")=((2-1)g)/((1+3+2))=(g)/(6)ms^(2)`
`T_(1)=m_(up)(g+a)=1(g+(g)/(6))=(7g)/(6)N`
`T_(2)=m_(down)(g-a)=2(g-(g)/(6))=(5g)/(3)N`.