Hooke s law wave equation pdf

You are still considering the same three particles you started. Because the force is proportional to displacement of the spring from its equilibrium position, a mass attached to the spring will undergo simple harmonic motion. The motion of the spring will be compared to motion of a pendulum. Following the derivation in the previous section on plane wave eigenmodes. We derive a wave equation for smallamplitude, undamped, extensional. Strain displacement equations in polar coordinates airy stress function in polar coordinates. Practice problems assess your knowledge of this mathematical formula, and the. In the theory of elasticity, hookes law is an approximation for certain. Materials for which hookes law is a useful approximation are known as linearelastic or hookean materials.

The key notion is that the restoring force due to tension on the string will be proportional 3nonlinear because we see umultiplied by x in the equation. The deforming force may be applied to a solid by stretching, compressing, squeezing, bending, or twisting. Applications of a recurring principle article pdf available in ajp advances in physiology education 334. Request pdf the extension and oscillation of a nonhookes law spring. The wave equation in the onedimensional case can be derived from hookes. It states that for a helical spring or any other elastic material, extension is directly proportional to the stretching force,provided elastic limit is not exceeded i. A more direct derivation of the formula for the speed of sound is rather. The extension and oscillation of a nonhookes law spring article in european journal of mechanics asolids 262. Where e is the proportionality constant and often referred as youngs modulus when nature of force is tensile. Hookes law in the diagram below is shown a block attached to a spring. Hooke s law relates the stretching force and extension produced. Derivation of the wave equation from hookes law generalization. Hooke s law hooke s law, f kx, where the applied force f equals a constant k times the displacement or change in length x.

Hookes law and simple harmonic motion rowan university. In position b a force f is used to compress the spring by a length equal to. Imagine an array of little weights of mass m are interconnected with mass less springs of length h and the springs have a stiffness of k. Fke, where k is the constant of proportionality called spring constant. The modern theory of elasticity is a generalized variation on hookes law, which states that the straindeformation of an elastic object or material is proportional to the stress applied to it. In position a the spring is at rest and no external force acts on the block. Derivation of 1d wave equation tu delft opencourseware. A brief overview of springs, hookes law, and elastic potential energy for algebrabased physics students many materials obey this law of elasticity as long as the load does not exceed the materials elastic limit. You are not changing anything other than writing the spring constant for one spring in another way. From slinkies to pinball, they bring us much joy, and now they will bring you even more joy, as they help you understand simple harmonic motion. The extension and oscillation of a nonhookes law spring request. Test your ability to define the spring constant and apply the hookes law formula. In this lab we will verify hooke s law and learn about simple harmonic motion. The wave equation is the consequence of another fundamental physical law.

In equation form, hookes law is fkx where f is the force needed, x is the distance the spring is stretched or compressed beyond its natural length, and k is a constant of proportionality called. Here we will derive the wave equation for homogeneous media, using the conservation of momentum newtons second law and the conservation of mass. Based on hookes law, the relationship between stress and strain is. The wave equation is an important secondorder linear partial differential equation for the. If we have an elastic material with a cylindrical cross section a, length l, which is affected by a force f, and therefore is compressedprolonged an amount x, hookes law states. This earlier work has shortcomings that we have overcome as pointed out in section 4.