Motion of two bodies - rough surface and rough pulley:

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

Figure shows two weights W₁ and W₂ connected by string, passing over rough pulley. The weight W₂ is placed on rough horizontal surface, while the weight W₁ is hanging free. Thus in this case force of friction will be acting on weight W₂ in opposite direction of motion. As string is passing over rough pulley. The tension on both sides of string will not be same.

Let,  μ₁ = Coefficient of Friction between Weight W2 and Horizontal plane  μ₂ = Coefficient of Friction between String and pulley T₁  = Tension in string to which weight W1  is attached

T₂  = Tension in the string to which weight W₂  is attached

Force of friction =  μ₁R₂  = μ₁W₂

Consider block W₁

W₁  - T₁ = (W₁/g).a                                                                                                                              ...(1)

Consider block W₂

T₂  -   μ₂.W₂ = (W₂/g).a                                                                                                                          ...(2)

Another equation is, T₁/T₂  = e ^μ.θ                                                                                                           ...(3)

Solve all the three equation for value of 'a', 'T₁' and 'T₂'