A system of four masses m_(1),m_(2),m_(3),m_(4),two springs and a fixed pulley is shown in figure. The system is kept at rest by attaching the lower thread to a rigid peg . Determine the acceleration of all the masses after the lower thread has been cut . The springs are massless, pulley is frictionless and massless.

A system of four masses m_(1),m_(2),m_(3),m_(4),two springs and a fixe

1.	A system of four masses m_(1),m_(2),m_(3),m_(4),two springs and a fixed pulley is shown in figure. The system is kept at rest by attaching the lower thread to a rigid peg . Determine the acceleration of all the masses after the lower thread has been cut . The springs are massless, pulley is frictionless and massless.
Answer» Solution :Clearly `(m_(1)+m_(2))gt(m_(3)+m_(4))` Let `T_(1)` be the tension in left spring and `T_(2)` that in right spring and T is tension in the string passing over the pulley. Then for equilibrium of system `T_(1)=m_(2)G,T=(m_(1)+m_(2))g`….(2) and `m_(3) g+T_(2)=T` From lastequation `T_(2)=T - m_(3)g=(m_(1)+m_(2)-m_(3))g`....(3) After the lower thread is cut off EQUATIONOF MOTION of mass `m_(1) ` is `m_(1)g+T_(1)-T=m_(2)a_(3)`....(4) Equation of motion of mass `m_(2)` is `m_(2)g-T_(1)=m_(2)a_(2)`....(5) Equation of motion mass `m_(3)` is `m_(3)g+T_(2)-T=m_(1)a_(1)`....(6) Equation of motion of mass `m_(4)` is `m_(4)g-T_(2)=m_(4)a_(4)`....(7) USING (1),(2) and (3) we GET `a_(1)=a_(2)=a_(3)=0` and `a_(4)=(m_(4)g-T_(2))/m_(4)=(m_(4)g-(m_(1)+m_(2)-m_(3)g)/m_(4)=((m_(3)+m_(4)-m_(1)-m_(2))g)/m_(4)`