Three rubber cords, each of force constant k are joined with...

Three rubber cords, each of force constant $k$ are joined with each other at one of their ends. The free ends of the cords are pulled maintaining them always in a plane. For a particular set of pulling forces an arrangement shown in the figure- $\mathrm{I}$ is obtained and for another set of pulling of forces arrangement shown in the figure-II is obtained. In the arrangement of figure-I, extended length of each cord is $L_1$; and in the arrangement of figure-II, extended lengths of each cord is $L_2>L_1$. (GRAPH CANT COPY) With the help of the given information, check validity of the following statements. (a) $l_{\mathrm{A}}=l_{\mathrm{B}}<l_{\mathrm{C}}$ and $k_{\mathrm{A}}=h_{\mathrm{B}}<k_{\mathrm{C}}$ (b) $l_{\mathrm{A}}=l_{\mathrm{B}}<l_{\mathrm{C}}$ and either $h_{\mathrm{A}}=h_{\mathrm{B}}>k_{\mathrm{D}}$ or $h_{\mathrm{A}}=h_{\mathrm{B}}<h_{\mathrm{C}}$ (c) $l_{\mathrm{A}}=l_{\mathrm{B}}<l_{\mathrm{C}}$ if $k_{\mathrm{A}}=k_{\mathrm{B}}>k_{\mathrm{c}}$ or $l_{\mathrm{A}}=l_{\mathrm{B}}>l_{\mathrm{C}}$ if $k_{\mathrm{A}}=k_{\mathrm{B}}<k_{\mathrm{C}}$ (d) $l_{\mathrm{A}}<l_{\mathrm{B}}<l_{\mathrm{C}}$ if $k_{\mathrm{A}}>k_{\mathrm{B}}>k_{\mathrm{C}}$ or $l_{\mathrm{A}}>l_{\mathrm{B}}>l_{\mathrm{C}}$ if $k_{\mathrm{A}}<k_{\mathrm{B}}<k_{\mathrm{C}}$

Three rubber cords, each of force constant $k$ are joined with each other at one of their ends. The free ends of the cords are pulled maintaining them always in a plane. For a particular set of pulling forces an arrangement shown in the figure- $\mathrm{I}$ is obtained and for another set of pulling of forces arrangement shown in the figure-II is obtained. In the arrangement of figure-I, extended length of each cord is $L_1$; and in the arrangement of figure-II, extended lengths of each cord is $L_2>L_1$.
(GRAPH CANT COPY)
With the help of the given information, check validity of the following statements.
(a) $l_{\mathrm{A}}=l_{\mathrm{B}}<l_{\mathrm{C}}$ and $k_{\mathrm{A}}=h_{\mathrm{B}}<k_{\mathrm{C}}$
(b) $l_{\mathrm{A}}=l_{\mathrm{B}}<l_{\mathrm{C}}$ and either $h_{\mathrm{A}}=h_{\mathrm{B}}>k_{\mathrm{D}}$ or $h_{\mathrm{A}}=h_{\mathrm{B}}<h_{\mathrm{C}}$
(c) $l_{\mathrm{A}}=l_{\mathrm{B}}<l_{\mathrm{C}}$ if $k_{\mathrm{A}}=k_{\mathrm{B}}>k_{\mathrm{c}}$ or $l_{\mathrm{A}}=l_{\mathrm{B}}>l_{\mathrm{C}}$ if $k_{\mathrm{A}}=k_{\mathrm{B}}<k_{\mathrm{C}}$
(d) $l_{\mathrm{A}}<l_{\mathrm{B}}<l_{\mathrm{C}}$ if $k_{\mathrm{A}}>k_{\mathrm{B}}>k_{\mathrm{C}}$ or $l_{\mathrm{A}}>l_{\mathrm{B}}>l_{\mathrm{C}}$ if $k_{\mathrm{A}}<k_{\mathrm{B}}<k_{\mathrm{C}}$

Key Concepts

Vector Addition and Force Geometry

When forces are applied in different directions, vector addition becomes essential to determine the resultant force. In systems where elastic cords are pulled in various directions, the directions and magnitudes of the forces must be considered via geometric construction (often forming closed polygons) to ensure equilibrium. This concept helps in understanding how different arrangements can satisfy the equilibrium conditions.

Dependence of Extension on Applied Force and Force Constant

The extension in an elastic cord is determined by both the magnitude of the applied force and the cord's force constant, as per Hooke’s Law. When external forces or the stiffness (k) of the cords vary, the resulting extensions also change. This dependency explains the differences in extension observed in various configurations of the system.

Hooke's Law

Hooke’s Law is fundamental in understanding elastic systems, stating that the force exerted by a spring (or an elastic cord) is directly proportional to its extension beyond its natural length. This proportionality is given by F = kx, where k represents the force constant and x the extension. This principle is crucial in analyzing how varying forces cause different extensions in elastic cords.

Equilibrium of Forces

The concept of equilibrium is central in mechanics, where an object or a system is in equilibrium if the net force acting on it is zero. In the context of multiple elastic cords attached at a common point, the vector sum of their tensions must cancel out. This ensures that the object remains at rest or moves with constant velocity, a critical idea in analyzing the stability of different configurations.

Multiple Equilibrium Configurations

In some mechanical systems, including those with elastic elements, there can be more than one configuration that satisfies the equilibrium conditions. This phenomenon of multiple stable or metastable equilibrium states means that under different sets of applied forces, the elastic cords may extend to different lengths while still balancing forces at the junction. Recognizing the possibility of multiple equilibria is key in understanding the behavior and stability of such systems.

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