Motion of two bodies - rough surface and smooth pulley:

THE HORIZONTAL SURFACE IS ROUGH AND THE STRING IS PASSING OVER SMOOTH PULLEY.

Figure shows two weights W₁ and W₂ connected by light inextensible string, passing over smooth pulley. The weight W₂  is placed on rough horizontal surface, while the weight W₁  is hanging free. Thus in this case force of friction is acting on weight W₂ in opposite direction of motion of weight W₂.

Let, μ  = The coefficient of friction between weight W2 and horizontal surface.
Force of friction =   R₂ = W₂

Motion of W₁  (Downward)

W₁  - T = (W₁/g).a                                                                                                                              ...(1)

Motion of W₂

T - μg W₂ = (W₂/g).a                                                                                                                           ...(2)

Solve equations for Tension 'T' and Acceleration 'a'