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write newton's second law for each mass

in doing this, remember that up is the positive direction, and down is negative

recall also that the tension in the string will be the same throughout

so for the mass accelerating upward, which I will call m, you have

T - m g = m a

for the larger mass which will descend, we have

T - M g = - Ma

both masses will have the same magnitude of acceleration, but the ascending one will be positive and the descending mass will have negative acceleration

if you solve these two equations simultaneously, you will get the value of a

a = (M-m)g/(m+M)

Suppose mass M is on the RHS and mass m is on the LHS.

T is the tension in the cord.

a is the common acceleration of the system.

Suppose mass M is moving downward and mass m is moving downward.

RHS :

Forces on the mass M are (i) weight Mg acting downward and (ii) Tension T acting upward.

Applying Newton's 2nd Law we get : Mg - T = Ma -------------- (1)

LHS :

Forces on the mass m are (i) Tension T acting upward and (ii) weight mg acting downward.

Applying Newton's 2nd Law we get : T - mg = ma ------------- (2)

Adding (1) and (2) we have : (M - m)g = (M + m) a

=> a = (M - m)g / (M + m) -------------- (3)

Subtracting (2) from (1) we have : (M + m)g - 2T = (M - m)a

=> 2T = (M + m)g - (M - m)a ; substituting the value of a from (3)

= (M + m)g - {(M - m)²g} / (M + m)

= {(M +m)² - (M - m)²}g / (M + m)

= 4Mmg / (M + m)

=> T = 2Mmg / (M + m) -------------- (4)