Particles of masses 2 M, m and M are respectively at points...

Particles of masses $2 M, m$ and $M$ are respectively at points $A, B$ and $C$ with $A B=\frac{1}{2}(B C) . m$ is much-much smaller than $M$ and at time $t=0$, they are all at rest (a) $m$ will remain at rest (b) $m$ will move towards $M$ (c) $m$ will move towards $2 M$ (d) $m$ will have oscillatory motion

Particles of masses $2 M, m$ and $M$ are respectively at points $A, B$ and $C$ with $A B=\frac{1}{2}(B C) . m$ is much-much smaller than $M$ and at time $t=0$, they are all at rest
(a) $m$ will remain at rest
(b) $m$ will move towards $M$
(c) $m$ will move towards $2 M$
(d) $m$ will have oscillatory motion

Key Concepts

Perturbation Analysis in Multi-Body Systems

Perturbation analysis involves studying the effects of small deviations from an exact equilibrium configuration. In systems with gravitational attractions, even a tiny imbalance can result in significant motion due to the nonlinear nature of gravitational forces. It is a key tool in determining the subsequent movement and stability of a body under nearly balanced forces.

Equilibrium and Stability

Equilibrium in a system occurs when the net force acting on a body is zero. However, not all equilibria are stable. In an unstable equilibrium, any slight deviation will result in forces that drive the system further away from equilibrium. This concept explains why a body positioned at a balance point might start moving if disturbed, an analysis that is central to dynamics in multi-body systems.

Gravitational Force

Gravitational force is the attractive force that exists between any two masses. It is proportional to the product of the masses and inversely proportional to the square of the distance between their centers. This concept is fundamental when analyzing interactions between bodies in any gravitational system.

Inverse-Square Law

The inverse-square law describes how the strength of the gravitational force diminishes with distance. Specifically, if the distance between two masses is doubled, the gravitational force becomes one fourth as strong. This principle is crucial for understanding how forces vary when bodies are positioned at different separations.

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