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## Homework Statement

a solid cylinder of mass m and radius R and a plank of mass M are placed on a smooth horizontal surface. there is sufficient friction between the cylinder and the plank's surface to prevent sliding of the cylinder. the fixed pulley is smooth. if the plank is pulled with a horizontal force F, the acceleration of center of the cylinder is? (Refer the figure *dia.bmp*),(take tension in the string as T and friction between the cylinder & plank as f).

View attachment dia.bmp

**2. The attempt at a solution**

1. started off with the force equation of the plank => F- ( T + f)= Ma ----equation(a)

2. equation of motion of cylinder => T-f = ma

_{cm}-----equation(1)

3. rotational equation of cylinder => fR = Iα

fR = mR

^{2}/2 * α

f = ma

_{cm}/2

substituting in equation(1) and solving => T = 3ma

_{cm}/2

we know, a

_{cm}= a/2 ( doubtful about the requisite)

so substituting the results in equation(a),

F - 2ma

_{cm}= 2Ma

_{cm}

=> a

_{cm}= F/2(M+m).

but the solution given the referred book is a

_{cm}= F/3m+M.

have i done some mistake is writing the equations or at some other point? please help! thanks.