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2 Blocks 2 Slopes

A common advanced problem involves 2 blocks attached by a rope over a pulley. Sometimes the pulley is massless, sometimes frictionless, and sometimes the angles are horizontal or vertical. The setup for this assumes the blocks are moving clockwise, if your problem is the other way, swap A's and B's. Instead of solving individual free-body diagrams for each object in both directions like in this example, we can only need one: the external forces in the direction of motion. In this case, the external forces are the x-components of the weights and the friction on both blocks. External forces like the normal force are balanced by the y-component of gravity, and internal forces like the tension in the rope are balanced by the other end of the rope. 


This solves the problem for acceleration, which can be used to find position and velocity as a function of time or distance using the kinematic equations. We use the "equivalent mass" of the pulley because different shapes have different moments of inertia. In short, the mass at the axle of the pulley is barely moving and the mass at the edge has to move at the same speed as the rope and the equivalent mass is a short way around solving the angular kinematics equations. The Work-Energy method can also be used to find a relationship between velocity and position. Because the length of the rope is constant, both blocks must move the same distance along the slope and have the same velocity as the edge of the pulley.


Friction is kept as shorthand "curly f" for space and the Rotational kinetic energy (RKE) is simplified with the equivalent mass. We can then solve for the distance traveled by the blocks or either velocity.

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