Force and torque heavy pulley Atwood machine - block slides down the ramp as the pulley unwinds.

In this problem, a block slides down a frictionless ramp, but the block is tied to a string wrapped around a heavy pulley. Our goal is to use force and torque analysis to find the acceleration of the block and tension in the string. We begin by labelling all the forces acting in the problem: the tension pulls on the block and the tension pulls at the edge of the heavy disk. Gravity pulls straight down on the block, and we only need the parallel component of the force of gravity, since the ramp is frictionless. We apply Newton's second law F_net=ma to the block: the net force is the component of gravity pointing down the hill minus the tension pulling up the hill, and this is equal to mass times acceleration for the block. Next, we apply the rotational form of Newton's second law: tau=I*alpha. The tension exerts a torque on the heavy disk given by TR since the string is perpendicular to the radius. The moment of inertia is 1/2*MR^2 for a disk, and the angular acceleration can be written in terms of the acceleration of the string using the angular to linear conversion a=r*alpha. We sub all the terms into tau=I*alpha, and we have the equation for the disk in the heavy pulley atwood machine. Now we solve the force and torque system of equations by using elimination to eliminate tension. We solve for a and obtain the acceleration of the block down the ramp as the pulley unwinds. Finally, we sub the acceleration into the torque equation and find the tension in the string for the heavy pulley and block problem.