Three Pulley Problems

The 4 kg mass will accelerate down with accelerationsay.If we assume the 2 kg mass also accelerates down, withaccelerationsay,the 1 kg mass will accelerate up with accelerationIfwe are wrong about the direction of any of the accelerations, thatacceleration will come out negative. With these assumption, we canlabel the diagram as below.

Applying

For the 2 kg mass: (1)

For the 1 kg mass: (2)

For the 4 kg mass: (3)

(1)-(3) gives (4)

2*(1) + (2) gives (5)

5*(4)+(5) gives

Substitute this into (4) to give

From (1)