Masses m and 3m are connected by a string, which is draped...

Masses $m$ and 3$m$ are connected by a string, which is draped over a pulley, as shown in the diagram above. The masses are released from rest. Determine the magnitude of acceleration of mass $m .$
(A) $\frac{g}{4}$
(B) $\frac{g}{3}$
(C) $\frac{g}{2}$
(D) $g$
(E) 2$g$

Key Concepts

Tension in a String and Ideal Pulley Assumption

When two masses are connected by a string over a pulley, it is typically assumed that the pulley is frictionless and the string is inextensible. These assumptions imply that the tension is the same throughout the string and that the accelerations of the connected masses are related, which simplifies the analysis of the system.

System of Equations and Constraint Relations

In systems where multiple objects are connected, the constraint that they move together (having the same magnitude of acceleration) allows us to set up a system of equations based on Newton's second law for each object. Solving these simultaneous equations is key to determining unknown quantities like the acceleration and tension.

Newton's Second Law

Newton's second law states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration (F = ma). This principle is applied to each object in a system to relate the forces acting on that object, such as gravitational force and tension, to its acceleration.

Free-Body Diagrams

A free-body diagram is a visual representation of all the forces acting on an object. In problems involving connected masses, drawing free-body diagrams for each mass helps isolate and correctly account for forces like gravity and tension, which are crucial for setting up the equations of motion.

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