As depicted in the figure, two blocks of masses m and M are linked by a string that passes over a pulley. The horizontal table on which the mass m slides is smooth. The pulley, which can freely rotate about its axis, has a radius r and a moment of inertia I about this axis. Determine the acceleration of the mass M , assuming the string does not slip on the pulley.

Step-by-step solution

Let a be the acceleration of the blocks and be the angular acceleration of the pulley. Since the string does not slip on the pulley, the relation between linear and angular acceleration is a = r . Let T₁ be the tension in the horizontal segment of the string attached to mass m , and T₂ be the tension in the vertical segment attached to mass M . Writing the equations of motion for each component: For mass M (moving downwards): Mg - T₂ = Ma For mass m (moving horizontally on a smooth table): T₁ = ma For the pulley (r