Hooke's law
Hooke's law
Hooke's law is an approximation of the response of elastic (i.e. spring-like) bodies. It states (in Latin): ut tensio, sic vis, or in modern English, extension is directly proportional to force. It can also be written mathematically as \[\Delta x\propto F\], where \[\Delta x\] is the extension/compression displacement of the elastic body and \[F\] is the applied force.
As an equation, it is written as \[F=-k \Delta x\], where \[k\] measures the stiffness of the spring. This is the initial equation stated by Robert Hooke in a paper in 1676, stating that the force required to extend a spring with stiffness of \[k\] was proportional to the distance, \[\Delta x\]. Later, the concept was extended to most solid bodies acting within their elastic range.
Additionally, we usually just use \[F=k\Delta x\] when we don't take the direction into account. The purpose of the negative can essentially be summed up as, the negative sign in this law serves to indicate that the direction of the restoring force and the change in length is in opposite directions.
Young's modulus
For a uniform rod or spring made of an elastic material, this stiffness is directly related to the material's Young's modulus \[E\] and inversely related to its natural length \[l\]. Young's modulus is defined as the ratio of stress to strain, i.e. \[E=\frac{\frac{F}{A}}{\frac{\Delta l}{l}}=\frac{Fl}{A\Delta l}\], where \[A\] is the cross-sectional area of the spring-like body and \[\Delta l\] is the change in length of the body. Rearranging this into an equation similar to Hooke's law (and writing \[\Delta x\] as \[\Delta l\]) gives us \[F=\frac{EA\Delta l}{l}=\frac{EA}{l}\cdot \Delta l\], then we can say that the spring constant \[k=\frac{EA}{l}\]. In simple scenarios where the area is not mentioned, we usually assume the cross-sectional area of the string \[A=1\], thus \[k=\frac{E}{l}\].
Sometimes we would denote the Young's modulus \[E\] as \[\lambda\], and generally refer to it as the modulus of elasticity. This gives us the final equation of \[F=k\Delta x=\frac{\lambda}{l}\Delta x\].
Conservation of energy
When an elastic body is compressed or extended, elastic potential energy will be stored in the system. When the elastic body is released, some or not all of the elastic potential energy will be converted into gravitational potential energy, \[\text{P.E.}=mgh\] and/or kinetic energy, \[\text{K.E.}=\frac{1}{2}mv^{2}\]. Elastic potential energy converts into gravitational potential energy when the elastic body recoils in an upwards direction, and it converts into kinetic potential energy when the elastic body is not at instantaneous rest.