Solve the Equilibrium of a Light Elastic String with Mass m

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SUMMARY

The discussion focuses on solving the equilibrium of a system involving two light elastic strings, AB and CD, with respective moduli of elasticity of 2mg and 4mg. The participant establishes the relationship between tensions T1 and T2, leading to the equation T2 - T1 = mg. The lengths and extensions of the strings are defined, with the total length AD expressed as AD = 2l + x + y. The participant seeks clarification on the missing distance between the fixed point and AD, indicating a need for further insights on the equilibrium conditions.

PREREQUISITES
  • Understanding of Hooke's Law and its application to elastic materials
  • Knowledge of equilibrium conditions in static systems
  • Familiarity with free-body diagrams and tension analysis
  • Basic principles of mechanics involving forces and mass
NEXT STEPS
  • Study the application of Hooke's Law in multi-string systems
  • Learn about free-body diagram techniques for analyzing forces
  • Explore equilibrium conditions in static mechanics
  • Investigate the effects of varying modulus of elasticity on string tension
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Students studying mechanics, particularly those focusing on elasticity and equilibrium in physics, as well as educators seeking to enhance their understanding of tension in elastic systems.

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Homework Statement




Hi
Can anyone please help with this question?
A light elastic string AB has natural length l and modulus of elasticity 2mg. Another light elastic string CD has natural length l and modulus of elasticity 4mg. The strings are joined at their ends B and C and the end A is attached to a fixed point. A particle of mass m is hung from the end D and is at rest in equilibrium. Find the length AD.

this is what I did

Let Tension AB is T2 and tension CD be T1. since system is at rest
T2-T1 = mg (Equation 1)
AD = l +x +y +l (Where x and y are the extensions of AB and CD respectively)
So, AD = 2l +x+y
Using Hookes law
T2 = 2mgx/l

Similarly, T1 = 4mgy/l

substituting the above into equation 1
2x-4y =l
This is as far as I can go. I wasn't given the distance between the the fixed point and AD. Am I missing something? Kindly comment.

Homework Equations





The Attempt at a Solution

 
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Draw a free-body diagram for the particle of mass m. What does the diagram tell you about the tension T1?

Also, what is your justification for claiming T2-T1 = mg ?
 

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