Two springs from the ceiling attached to one mass, least energy principle

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SUMMARY

The discussion focuses on a physics problem involving a mass connected to two springs with constants k and K, attached to the ceiling at a distance d apart. The objective is to determine the equilibrium coordinates (x,y) of the mass using energy minimization methods, specifically applying the least energy principle. The potential energy function is defined as φ=1/2k(stretch)^2, and the equilibrium condition is established by setting the derivative ∂φ/∂r to zero. The challenge lies in accounting for the simultaneous changes in the lengths of both springs and their angles with the ceiling.

PREREQUISITES
  • Understanding of Hooke's Law and spring constants (k and K).
  • Familiarity with energy minimization techniques in physics.
  • Knowledge of potential energy functions and their derivatives.
  • Basic concepts of vector mathematics and coordinate systems (Cartesian and polar).
NEXT STEPS
  • Study the application of the least energy principle in mechanical systems.
  • Learn about the derivation and application of potential energy functions in multi-spring systems.
  • Research methods for solving systems of equations involving multiple variables and constraints.
  • Explore the use of Lagrange multipliers for handling constraints in physics problems.
USEFUL FOR

Physics students, mechanical engineers, and anyone interested in understanding the dynamics of spring systems and energy minimization techniques.

charliepebs
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Suppose have a ball connected to a spring on each end – one with constant k, and the other with constant K. And suppose the springs are attached to the ceiling a distance d apart. Use energy minimization methods to determine the (x,y) coordinates of the mass in equilibrium – taking the origin of the coordinate system to be the point where spring K is attached to the ceiling.





2. φ=1/2k(stretch)^2
least energy principle: ∂φ/∂r=0

3. Not a lot of specifics given in this problem as far as initial conditions, stretched upon attachment, etc., not sure whether to use polar or Cartesian. I know I need a constraint, probably that since the springs are connected to the same point, the vector of one plus the vector of the other must equal the distance between them. Just not sure how to handle the fact that length of both springs and the angle they both make with the ceiling is changing simultaneously.
 
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Length of each spring? Obviously matters. Or can the answer be given in terms of the lengths L1 and L2 so that the mass m is situated at p(x,y), x= x(m,L1, L2, d) and y = y(m,L1,L2,d)?
 
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