I think that if you are getting to a point where the spring loop diameter(s) or thickness(es) of wire segment(s) are diverging from initial values, then you need a new model... Hooke's Law is a fundamental principle in physics that describes the relationship between the force applied to a spring and its resulting deformation. The law states that the force (F) required to stretch or compress a spring by a certain distance (x) is proportional to that distance, with a constant of proportionality known as the spring constant (k). Mathematically, this can be expressed as: F = -kx where the negative sign indicates that the force is opposite in direction to the displacement. Spring Loop Diameters The diameter of a spring loop refers to the distance between the center of the spring and its outermost point. The diameter of a spring loop can affect the spring's behavior and its ability to follow Hooke's Law. Here are some key points to consider: Constant diameter springs: Springs with a constant diameter will have a consistent spring constant (k) and will follow Hooke's Law over a wide range of displacements. Variable diameter springs: Springs with a variable diameter, such as conical or convex springs, will not follow Hooke's Law over a wide range of displacements. The spring constant (k) will vary depending on the diameter of the spring. Elastic limit: The elastic limit of a spring refers to the maximum amount of deformation it can withstand before it becomes permanently deformed. The elastic limit is affected by the diameter of the spring loop, with larger diameters typically resulting in a higher elastic limit. Relationship between Spring Loop Diameters and Hooke's Law The relationship between spring loop diameters and Hooke's Law is complex and depends on the specific type of spring and its material properties. However, in general, the following can be said: Linear springs: Springs with a constant diameter will follow Hooke's Law over a wide range of displacements, with the spring constant (k) remaining constant. Non-linear springs: Springs with a variable diameter will not follow Hooke's Law over a wide range of displacements, with the spring constant (k) varying depending on the diameter of the spring. Elastic limit: The elastic limit of a spring is affected by the diameter of the spring loop, with larger diameters typically resulting in a higher elastic limit. In conclusion, the diameter of a spring loop can affect the spring's behavior and its ability to follow Hooke's Law. While constant diameter springs will follow Hooke's Law over a wide range of displacements, variable diameter springs will not. The elastic limit of a spring is also affected by the diameter of the spring loop, with larger diameters typically resulting in a higher elastic limit.
The spring constant (k) is a measure of the spring's stiffness, and it is typically dependent on the spring's material, shape, and size. The loop diameter of a spring is one of the factors that can affect the spring constant. Effect of Loop Diameter on Spring Constant As the loop diameter of a spring increases, the spring constant (k) typically decreases. This is because a larger loop diameter means that the spring has more flexibility and can stretch further before reaching its elastic limit. As a result, the spring requires less force to achieve a given deformation, which means that the spring constant is lower. Conversely, as the loop diameter of a spring decreases, the spring constant (k) typically increases. This is because a smaller loop diameter means that the spring has less flexibility and can only stretch a shorter distance before reaching its elastic limit. As a result, the spring requires more force to achieve a given deformation, which means that the spring constant is higher. Mathematical Representation The relationship between the loop diameter and the spring constant can be represented mathematically as: k ∝ 1/diameter^2 where k is the spring constant, and diameter is the loop diameter of the spring. This equation suggests that as the loop diameter increases, the spring constant decreases, and as the loop diameter decreases, the spring constant increases. Conclusion In summary, varying the loop diameter of a spring can affect the spring constant, which is a critical component of Hooke's Law. A larger loop diameter typically results in a lower spring constant, while a smaller loop diameter results in a higher spring constant. This relationship can be represented mathematically as k ∝ 1/diameter^2.
It looks like the spring when compressed is just stopped by its own coils. How much force does it take to move the spring at all? The "real" zero point isn't the length of that spring I think, it stores tension at rest.
I think if you put a weight on it so it hung loosely, then used that as the zero point, then relative to that point you'd see double extensions when hanging the one and double weight. The spring constant isn't really constant, but it's a lot closer than this.
Limitations of Hooke's Law Hooke’s Law, which states that the stress in a material is proportional to its strain within the elastic limit, has several limitations. These limitations are: Elastic Limit: Hooke’s Law is only applicable within the elastic limit of a material. Beyond this point, the material becomes plastic and Hooke’s Law no longer holds. Material Properties: Hooke’s Law is only applicable to materials that exhibit linear elastic behavior. Many materials, such as metals, exhibit non-linear behavior, making Hooke’s Law inapplicable. Deformation: Hooke’s Law is only applicable for small deformations. As the deformation increases, the material’s behavior becomes non-linear and Hooke’s Law is no longer valid. Temperature: Hooke’s Law is temperature-dependent. The elastic properties of a material change with temperature, making Hooke’s Law inapplicable at high or low temperatures. Non-Uniform Stress: Hooke’s Law assumes uniform stress distribution. In reality, stress distribution is often non-uniform, making Hooke’s Law inapplicable. Time-Dependent Behavior: Hooke’s Law assumes instantaneous response to stress. In reality, materials exhibit time-dependent behavior, making Hooke’s Law inapplicable for dynamic loading conditions. Microstructural Effects: Hooke’s Law neglects microstructural effects, such as grain size, defects, and impurities, which can affect a material’s elastic behavior. Non-Elastic Behavior: Hooke’s Law is only applicable to elastic behavior. Materials can exhibit non-elastic behavior, such as plastic deformation, creep, and fatigue, which are not accounted for by Hooke’s Law. These limitations highlight the importance of considering the specific material properties and loading conditions when applying Hooke’s Law.
Deriving Hooke's law for a metallic spring from the forces between its particles is a complex process that involves understanding the molecular structure and behavior of metals. At the atomic level, a metal can be considered as a lattice of atoms connected by interatomic bonds. When a metal is *deformed, the interatomic bonds get stretched, leading to a restoring force. *(For short deformations). This restoring force is proportional to the amount of deformation, which is the essence of Hook's law. However, deriving the exact mathematical formula (F = -kx) from the interatomic forces is a complex task that requires a deep understanding of quantum mechanics and solid-state physics. It's typically done in the field of materials science.
Coulomb's law describes the electrostatic force between charged particles, which isn't directly applicable to the forces between particles in a metallic spring, as metals are held together by metallic bonds, not electrostatic forces. However, the principle of Hooke's law can be derived from the behavior of springs at the macroscopic level, considering the deformation of the material when a force is applied, without needing to delve into the atomic level. This is typically done using the framework of continuum mechanics.
Continuum mechanics is a branch of physics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. It's not directly related to the behavior of individual springs, which are typically modeled using simpler theories like Hooke's law. However, continuum mechanics can be used to model the behavior of large collections of springs or other elastic materials under more complex conditions.
In the context of continuum mechanics, a spring can be modeled as a one-dimensional body. This model assumes that the spring is a continuous material, rather than a collection of discrete particles. The stress-strain relationship in a spring can be described by Hooke's law, which is a simple linear relationship between the stress (force per unit area) and strain (partial derivative of the displacement with respect to the spatial coordinates). This model allows for the analysis of the spring's behavior under different conditions, such as varying loads or temperatures, while treating the spring as a continuous material.
In the context of continuum mechanics, a spring can be modeled as a one-dimensional body. The stress (σ) in a spring can be described by Hooke's Law, which in general terms is σ = Eϵ, where E is the Young's modulus (a measure of the material's stiffness) and ϵ is the strain. The strain is defined as the partial derivative of the displacement (u) with respect to the spatial coordinate (x), or ϵ = ∂u/∂x. This model assumes that the spring is made of a homogeneous, isotropic material and that the deformation is small.
