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Rigid body rotation and movement

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I am experimenting with a 2D physics simulation regarding rigid bodies,
Given the situation described in the picture:
1) A resting rigid body of: Mass of M, Angular Mass of I, and a length of 2r (a radius of r). (The mass is elongated and completely flat)
2) Two forces at both ends perpendicular to the mass.

I need to figure out the torque and acceleration of said mass.
Integrating the torque is easy (just sum up the sum of torques generated by the forces)

But now comes the tricky part... How do I calculate the linear acceleration of the mass (the rate at which the centre is accelrating)?

Intuitively I think it's a=2*(0.5 F)/M or a=F/M (just deriving this from pulley and tension physics I remember from high-school)

But also, I need a clear algorithm (not just intuition) to figure out the mass's accelration.
Also, In the real simulation there would be more than two forces acting on the mass.

I'm thinking maybe I can calculate this via conservation of momentum: After summing up the torques and forces:
Accelerating_Force_Of_Center_Of_Mass = Sum_Of_Forces_Applied * (1 - |Sum_Of_Resulting_Tourques| )

So how do I find the acceleration of the centre of the mass?

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Actually, the linear accelleration is just the sum of all linear forces divided by mass :) so in your example, it's just 1.5F/M downwards, it doesn't matter where these forces are applied the linear acceleration is still the same.

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