Young's modulus & Hooke's law (AI generated article) Young’s modulus and Hooke’s law are two fundamental concepts in the field of materials science and engineering. They are closely related and describe the behavior of materials under stress and strain. What is Young’s Modulus? Young’s modulus is a measure of a material’s stiffness or resistance to elastic deformation under load. It is defined as the ratio of stress (force per unit area) to strain (proportional deformation) within the proportional limit of the material. Mathematically, it is represented as: E = σ / ε where E (sometimes "Y" is used) is the Young’s modulus, σ is the stress, and ε is the strain. What is Hooke’s Law? Hooke’s law is a law of physics that describes the relationship between stress and strain in a material. It states that the stress (force per unit area) is proportional to the strain (proportional deformation) within the proportional limit of the material. Mathematically, it is represented as: F = kx where F is the force, k is the spring constant, and x is the displacement from the equilibrium position. Relationship between Young’s Modulus and Hooke’s Law Young’s modulus and Hooke’s law are closely related. In fact, Young’s modulus can be derived from Hooke’s law. The spring constant (k) in Hooke’s law is related to the Young’s modulus (E) by the following equation: k = EA/L where L is the length of the material. Key Takeaways Young’s modulus is a measure of a material’s stiffness or resistance to elastic deformation under load. Hooke’s law describes the relationship between stress and strain in a material. Young’s modulus can be derived from Hooke’s law, and the two concepts are closely related. The higher the Young’s modulus, the more stress is needed to create a given amount of strain. Examples: A metal rod with a high Young’s modulus will require more force to stretch it a given distance than a rod with a lower Young’s modulus. A spring with a high spring constant (k) will require more force to compress it a given distance than a spring with a lower spring constant. In summary, Young’s modulus and Hooke’s law are fundamental concepts in materials science and engineering that describe the behavior of materials under stress and strain. They are closely related and are used to predict the elastic properties of materials.
Perfect, Ian! The force (F) gets higher with the stretch (x): F = kx The energy (E) gets higher with the stretch (x): E = Fx That is x squared, thus! E = ½kx² 👍
@@_John_Sean_Walker hi John, yes Ian has made a clear and concise video demonstrating a logical analysis of the forces involved versus spring expansion. The question I have for you is how do you explain where and how is the energy stored? Is it internal to the spring alone or are there other factors.
@@postiemania Force vector diagrams can get very complicated, think alone of the diagram of throwing a tennis ball against a wall, your hands are above your feet, so there must be angles at work, and when your feet don't slip, they are through the Earth connected with the wall, but in Ian's experiment we can say that the wall doesn't contribute to the energy that is released when you let go of the spring, but without the wall there couldn't be any tension on the spring, and don't forget that it is Ian's hand that guides all the potential spring energy into the spring, and that pulls on his body too. But in general we can say that it is the spring that stores the energy, yes. You can compare it with a wooden bow, in the end, it are the electrons that hold the wood together.
How about Young's Modulus❔ E=Stress/Strain= (F/A)/(∆x/l) => F= (E·A/l)∆x. E~ const, A/l~❔🤔 Does the ratio of wire crossectional area to spring wire length vary❔
Nonsense, has nothing to do with it. What Ian shows is: (Work is a scalar quantity.) How to get from: F = kx to: E = ½kx² The work *_𝑊_* done on the spring is equal to the energy stored in the spring (Elastic Potential Energy), which is the integral of the force over the displacement: W=∫Fdx W=∫kx dx W=∫0..x kx dx W=k∫0..x x dx from: x•dx to: [x²/2] = from: 0 to x = ½x (in triangle) W=k[x²/2]0..x W=k(x²/2) - k(0²/2) W=k(x²/2) W=½kx² The step (4) from: W=k∫ to: W=k[ is called the Power Rule.
@@_John_Sean_WalkerI know all that. Trivial. Also, F and dx are _vectors_ , so you should be integrating the dot product F•dx in general. If you check out my comments on his "Hookes law fail?" video, I list all the limitations of Hookes law and show alternative models.
@@The_Green_Man_OAP Wrong again! You missed the sentence: "How to get from: F = kx to: E = ½kx²" This is important, because otherwise you might think it would be: F = kx and E = Fx thus E = kx² But that is wrong, and it took quite some time to assemble the derivation I showed, and you think that I'm just rumbling around, but the derivation is in *ALL* the books. This stuff goes back to Émilie du Châtelet and Leonhard Euler! I also wrote right at the beginning: "Work is a scalar quantity." but you missed that too, because you start nagging about vectors, while the video is about *work.* You should have said: "Wow Ian, beautiful video, dude!" All but trivial!
@@The_Green_Man_OAP No, dx is not a vector like F. Here’s a detailed explanation of the differences: dx in Calculus 1. Meaning: - In calculus, dx represents an infinitesimally small change or increment in the variable x. It is used in the context of integration and differentiation. - When you see dx in an integral ∫ f(x) dx, it signifies that you are integrating the function f(x) with respect to x. 2. Nature: - dx is a differential element, not a vector. It is a scalar quantity representing a very small change in x. F as a Vector 1. Meaning: - F typically represents a force vector in physics. A force vector has both magnitude and direction. - For example, in three-dimensional space, a force vector can be represented as *F* = (F_x, F_y, F_z), where F_x, F_y, and F_z are the components of the force in the x, y, and z directions, respectively. 2. Nature: - F is a vector quantity. Vectors have both magnitude and direction and are often represented by arrows in graphical depictions. Key Differences - Dimensionality: dx is a scalar, while F is a vector. - Context: dx is used in calculus to denote an infinitesimal change in a variable, while F in physics denotes a force, which is inherently a vector. Visualization - Scalar: A scalar (like dx) can be thought of as just a number, representing size or magnitude without direction. - Vector: A vector (like F) has both a size (magnitude) and a specific direction. Example in Context 1. Integral Example: - When integrating a function f(x): ∫ f(x) dx dx indicates that the integration is with respect to x. 2. Vector Force Example: - A force vector *F* acting on an object: *F* = (3 N, -2 N, 5 N) This vector has components in the x, y, and z directions. In summary, dx is a differential element representing an infinitesimal change in a variable, while _F_ is a vector representing force with both magnitude and direction. They are fundamentally different concepts used in different contexts.
