Pulley on a double inclined plane

A classic case: two masses connected by a rope passing over a pulley at the top of a triangular wedge, with the two bodies on differently inclined sides. It is the testing ground for the general method of constrained systems.

The strategy exploits the properties of the ideal rope and pulley (see Properties of the ideal rope): the two bodies have the same acceleration in magnitude $a$ and are subject to the same tension $t$. Newton’s second law is then written, along the respective plane, for each body separately. This gives two equations in the two unknowns $a$ and $t$.

The trick to solving it is to add (or subtract) the two equations written with a consistent sign convention along the rope: this way the tension, which appears with opposite signs in the two bodies, cancels out, leaving a single equation in the acceleration alone. Once $a$ is found, it is substituted back into one of the two to obtain $t$.

Method — Ideal pulleys

For ideal pulleys:

• Same $|a|$ at the two ends of the rope.
• Same tension $t$.
• Write the second law for each body, then add to eliminate $t$.

The complete numerical application (two masses on a plane at $60^\circ$ and $30^\circ$, with a spring acting on one of them) is worked out in Two masses on a triangular plane.

Links

Topics: Dinamica Concepts: Seconda legge di Newton · Tensione Skills: Applicazione delle leggi di Newton · Impostazione simbolica Objects: Piano inclinato · Puleggia · Fune ideale

Related exercises: Inclined plane with pulley and hanging mass · The Atwood pulley: the force on the axle · Two masses on a triangular plane