See attached picture. The acceleration of the plank is ap. F-2Fp=Map (1)
The motion of the cylinder is translation of the centre of mass and rotation about the CM.
The CM accelerates with acm. The friction between cylinder and plank accelerates the CM of the cylinders forward, and accelerates forward rotation; the rotational resistance Fg between ground and cylinder drives translation forwards, but hinders rotation. .
So Fp+Fg=macm. (2)
r(Fp-Fg)=Iβ (β is the angular acceleration)
The cylinders do not slip: βr=0.5 m.
The moment of inertia about the CM is I=0.5 mr2.
The torque equation becomes:
Adding equations (2) and(3):
The plank does not slip on the cylinders, so its velocity is the same as the topmost point of the cylinders, which is twice the velocity of the CM:
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