A trolley T (mass 5 kg ) on a horizontal smooth surface is...

A trolley $T$ (mass $5 \mathrm{~kg}$ ) on a horizontal smooth surface is pulled by a load $L(2 \mathrm{~kg})$ through a uniform rope $A B C$ of length $2 \mathrm{~m}$ and mass $1 \mathrm{~kg} .$ As the load falls from $B C=0$ to $B C=2 \mathrm{~m}$, its acceleration (in $\mathrm{ms}^{-2}$ ) changes from (Take $g=10 \mathrm{~ms}^{-2}$ ) (a) $20 / 6$ to $20 / 5$ (b) $20 / 8$ to $30 / 8$ (c) $20 / 5$ to $30 / 6$ (d) None of these

A trolley $T$ (mass $5 \mathrm{~kg}$ ) on a horizontal smooth surface is pulled by a load $L(2 \mathrm{~kg})$ through a uniform rope $A B C$ of length $2 \mathrm{~m}$ and mass $1 \mathrm{~kg} .$ As the load falls from $B C=0$ to $B C=2 \mathrm{~m}$, its acceleration (in $\mathrm{ms}^{-2}$ ) changes from (Take $g=10 \mathrm{~ms}^{-2}$ )
(a) $20 / 6$ to $20 / 5$
(b) $20 / 8$ to $30 / 8$
(c) $20 / 5$ to $30 / 6$
(d) None of these

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Key Concepts

Variable Geometry in Mechanisms

In scenarios where the configuration of a system changes during motion—such as a rope transitioning from a slack to a fully extended position—the effective distribution of mass and the resulting tension forces change. This leads to variations in acceleration as the system geometry evolves over time.

Dynamics of Connected Bodies

In systems where objects, like a trolley and a falling load, are linked through a connection (rope, pulley, etc.), the motion of one object directly influences the motion of the other. Analyzing such systems involves careful consideration of tensions, forces, and the interrelationship of the components’ accelerations.

Newton's Second Law

Newton's second law relates the net force acting on an object to its mass and acceleration. This principle is fundamental in analyzing any dynamic system, as it allows the determination of acceleration when the forces and masses of the various components are known.

Effective Mass in Systems

When connected bodies include elements with non-negligible mass, such as a rope or cable, the mass distribution must be accounted for as it contributes to the overall inertia of the system. This alters the effective mass that accelerates under a given force, affecting the system's acceleration.

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