Two weights w1 and w2 are suspended from the ends of a light...

Two weights $w_{1}$ and $w_{2}$ are suspended from the ends of a light string over a smooth fixed pulley. If the pulley is pulled up with acceleration $g$, the tension in the string will be (a) $\frac{4 w w_{2}}{w_{1}+w_{2}}$ (b) $\frac{2 w w_{2}}{w_{1}+w_{2}}$ (c) $\frac{w_{1}-w_{2}}{w_{1}+w_{j}}$ (d) $\frac{w_{1} w_{2}}{2\left(w_{1}+w_{2}\right)}$

Two weights $w_{1}$ and $w_{2}$ are suspended from the ends of a light string over a smooth fixed pulley. If the pulley is pulled up with acceleration $g$, the tension in the string will be
(a) $\frac{4 w w_{2}}{w_{1}+w_{2}}$
(b) $\frac{2 w w_{2}}{w_{1}+w_{2}}$
(c) $\frac{w_{1}-w_{2}}{w_{1}+w_{j}}$
(d) $\frac{w_{1} w_{2}}{2\left(w_{1}+w_{2}\right)}$

Key Concepts

Newton's Second Law

This fundamental principle states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration. In problems involving pulleys and accelerated systems, Newton's Second Law is applied separately to each mass to set up equations of motion that account for the various forces acting on them, such as gravitational force and tension.

Non-Inertial Reference Frames

A non?inertial (accelerating) reference frame is one in which objects appear to experience fictitious forces due to the acceleration of the frame itself. When solving problems where the pulley is accelerating, it is often useful to view the system from a frame attached to the accelerating pulley, incorporating pseudo forces to account for the acceleration.

Effective Gravity

In a non?inertial frame, the real gravitational acceleration is coupled with the acceleration of the frame to produce an effective gravitational field. This effective gravity alters the net forces acting on the masses, and it must be correctly accounted for when determining the acceleration of the masses and the tension in the string.

Constraint Relations in Pulley Systems

Pulleys impose geometric constraints on the motion of connected masses, such that the accelerations of the masses are related in specific ways. Recognizing and applying these constraint relationships is essential for linking the motion of different parts of the system, thereby enabling the solution of the simultaneous equations that yield the system's dynamics.

Tension in a String

The tension is the internal force transmitted through a string or rope that connects masses in a pulley system. Determining the tension requires careful analysis of the forces acting on each mass, including inertial, gravitational, and pseudo forces (when in an accelerated frame), and applying Newton's Second Law to each mass while considering the constraints imposed by the string.

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