Two blocks of masses m and 2 m are connected by a light...

Two blocks of masses $m$ and $2 m$ are connected by a light string passing over a frictionless pulley. As shown in the figure, the mass $m$ is placed on a smooth inclined plane of inclination $30^{\circ}$ and $2 \mathrm{~m}$ hangs vertically. If the system is released, the blocks move with an acceleration equal to (a) $\frac{g}{4}$ (b) $\frac{g}{3}$ (c) $\frac{g}{2}$ (d) $g$

Two blocks of masses $m$ and $2 m$ are connected by a light string passing over a frictionless pulley. As shown in the figure, the mass $m$ is placed on a smooth inclined plane of inclination $30^{\circ}$ and $2 \mathrm{~m}$ hangs vertically. If the system is released, the blocks move with an acceleration equal to
(a) $\frac{g}{4}$
(b) $\frac{g}{3}$
(c) $\frac{g}{2}$
(d) $g$

Key Concepts

Simultaneous Equations in Dynamics

In problems with multiple connected bodies, setting up and solving simultaneous equations based on the forces acting on each mass is necessary. This combined approach allows for the determination of unknowns such as acceleration and tension by ensuring consistency across the entire system.

Pulley Systems and Tension Uniformity

In ideal pulley systems with light strings and frictionless pulleys, the tension remains constant throughout the string. This assumption simplifies the analysis by allowing the equations of motion for connected masses to be linked together through the shared tension.

Free-body Diagrams

Drawing free-body diagrams helps in identifying and resolving all forces acting on an object. For systems involving inclined planes and hanging masses, these diagrams ensure that gravitational forces, tensions, and normal forces are correctly represented and resolved along appropriate directions.

Resolving Forces on an Inclined Plane

When an object is on an inclined plane, its weight must be resolved into components parallel and perpendicular to the plane. The parallel component, which is the product of the weight and the sine of the inclination angle, is critical in determining the motion along the plane.

Newton's Second Law

Newton's second law, F = ma, is fundamental for setting up the equations of motion for each block. By equating the net force acting on each mass to the product of its mass and acceleration, one can derive relationships between forces such as gravity and tension in the string.

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