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

In summary, the conversation discusses using energy minimization methods to determine the equilibrium coordinates of a mass suspended by two springs with different constants and attached to the ceiling a distance apart. The principle of least energy is mentioned, along with the need for a constraint due to the springs being connected at the same point. The question of whether to use polar or Cartesian coordinates is raised, as well as the impact of the lengths of the springs on the coordinates of the mass.
  • #1
charliepebs
7
0
1.
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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  • #2
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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Related to Two springs from the ceiling attached to one mass, least energy principle

1. How does attaching two springs from the ceiling to one mass relate to the least energy principle?

The least energy principle states that a system will always try to minimize its energy or find a state of equilibrium. When two springs are attached to one mass, the system will adjust itself to a position where the potential energy is at its minimum, following the least energy principle.

2. How do the properties of the springs affect the behavior of the system?

The properties of the springs, such as the spring constant and the length, determine the amount of force that is exerted on the mass and the rate at which the potential energy changes. These properties play a crucial role in determining the position of the mass and how it responds to external forces.

3. What happens when one of the springs is stiffer than the other?

If one of the springs is stiffer than the other, it will exert a greater force on the mass and cause it to move towards that spring. The system will then adjust itself until a state of equilibrium is reached, with the mass being closer to the stiffer spring.

4. Can the position of the mass be changed by adjusting the properties of the springs?

Yes, the position of the mass can be changed by adjusting the properties of the springs. For example, if the spring constant of one spring is increased, it will exert a greater force on the mass and cause it to move towards that spring. This will change the position of the mass in the system.

5. Is the least energy principle applicable to other systems in physics?

Yes, the least energy principle is a fundamental concept in physics and applies to many systems, such as pendulums, magnets, and electric circuits. It is a principle that governs the behavior of many physical systems and is essential for understanding and predicting their behavior.

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