This time one of the weights is placed on a slope. The pulley sits at the top of the slope, the string runs parallel to the surface, and the other weight hangs straight down.
What has changed is the force on the holding side. Hanging straight down, the whole weight pulled on the string; on a slope, only the part along the surface counts.
Split gravity into a component along the surface and one perpendicular to it. Along the surface it is , perpendicular to it . At zero tilt the component along the surface is ; stood vertical, it becomes itself.
The perpendicular component is what the surface pushes back against, so the normal force is . This is what fixes the limit of friction, so it is worth having in hand first.
The hanging weight pulls the string with . The holding side offers . The difference between them is the force trying to set things moving.
A difference does not start the motion at once, because friction takes up the rest. Static friction comes out to match whatever it is given and has no size of its own.
It has a limit, though: the maximum static friction . With , that limit is .
The figure makes the hanging weight heavier and heavier. The friction arrow grows to hold the balance and stops when it reaches the marker for the limit. Past that point it can hold no longer.
Write the condition for starting to move: the driving force must exceed the limit of friction, that is, .
Rearranged, this is , and dividing both sides by gives .
The acceleration due to gravity has dropped out. Whether the system moves is fixed by the ratio of the two masses, the tilt and the coefficient of friction, and nothing else. Carry the whole apparatus to the Moon and the threshold is unchanged.
Set the tilt to and it reads , the story of a load on a level table pulled by a string. Remove the friction and it reads , the story of a tilt letting a lighter weight suffice. Slope and level surface both sit at the ends of this one inequality.
Past the threshold the system moves, and the friction becomes kinetic friction. Its size is , constant and independent of speed. It does not adjust itself to its partner the way static friction did.
From here the two are treated as one body. The driving force is , the holding forces are and , and the mass being driven is .
Together they give . Every term on the right is a fixed value, so is constant, and the motion after the start is uniform acceleration.
Set this beside the Atwood equation. Two terms, and , have been added, and nothing else about the form has changed. Divide the driving force by the total mass: the method is the same in both.