#mainpage --- ## Springs - A force that restores a system to an **equilibrium position is called a restoring force.** Systems that exhibit such restoring forces are **called elastic systems**. - The **magnitude of the spring force** is ==**proportional to the distance of the end of the spring from its equilibrium position**==. This means that the more you stretch a string, the more **it will resist that stretching** *(this also applies to pushing a spring inwards)*. Since this is a *linear* relationship we can easily model it using an equation. - Note that the following equation - **known as Hooke's law**, can only be used to calculate forces along one axis, either x or y *(in other words it assumes you are stretching/compressing the elastic object along just one axis)*. The negative sign in front of the k denotes the fact that the restoring force will always be applied in **the direction opposite** of the displacement. $\text{Hooke's law:}$ $\Huge\boxed{\;\large\color{LightCyan}F_{sp\,x}=-k\,\Delta x\;\;\;\;\;\;\;\;\;k=\frac{\Delta F}{\Delta x}\;}$ $\newline$ - The variable $\small k$ in this equations **depends on the strength of the spring**. If this value is large, that indicates that the corresponding spring *(or elastic object)* is stiff and if the value is small the string is weak. We call this value the **spring constant**. To find this value you must find the relationship between the length of the string's displacement $x$ and the force that the spring imparts $y$. - This value also corresponds to the **slope of a graph that shows the displacement $\small\Delta l$ vs the restoring force $\small F$** that an elastic object exerts. - The **spring constant is always measured in units of force per distance** *(This usually ends up being: newton's per meter: $\small N/m$).* --- ## Stretching and compressing materials - Since all materials impart a **restoring force** when being stretched or compressed like a spring, we can **adapt the above equations** to find the *spring constant* of a solid object. Since the **material being deformed greatly impacts the strength of the restoring force an additional variable $\small Y$,** known as ==*Young's modulus*== must be added that accounts for the **type of material that is being deformed**. The variable $A$ denotes the cross sectional surface area of the object being deformed and the variable $L$ denotes it's length.* - Materials that **strongly** resist being stretched, *like steel* are known as **rigid materials**. - Materials that can undergo large **deformations while exerting only small restoring forces are known as *pliant materials*. $\text{Spring constant for a solid material: }\;\;\huge\boxed{\normalsize\color{LightCyan}k=\frac{Y\cdot A}{L}}$ $\newline$ >[!quote] >##### *Visual of elastic material* >![[Springs and elasticityspringforce.png|850]] - Now that we have an equation for the spring constant $k$ of a solid object we can adapt **Hooke's law** to model the restoring force imparted by an *object* when it is deformed. - The **force per cross sectional area** is known as the **stress** and the ratio between the displacement and the objects **original length** is known as the **strain**. - You can find the stress of an object by multiplying its strain by $Y$ and you can find the strain of an object by dividing its stress by $Y$. - If the stress on an object **is due to it stretching** we call it a **tensile stress**. - Strain is measured in units such as **$\small \frac{in}{in}$ or $\small\frac{mm}{mm}$,** these units are referred to as inch per total inch or millimeter per total millimeter *(unit per total unit)*. You will also commonly hear the unit *"microstrain"* which refers to $\small 1\cdot 10^{-6}$ millimeters of strain. - Stress is measured in units of **force per area**. *For example, $\small N/m^2$ or $\small lb/in^2$.* This unit is also commonly referred to **as a pascal (pa)**. $\newline$ $\text{Hooke's law for solid objects and stress/strain:}$ $\Huge\boxed{\normalsize\color{LightCyan}F=\frac{Y\cdot A}{L}\cdot\Delta L\;\;\;\;\;\;\;\;\;\;\;\text{Stress: }\frac{F}{A}\;\;\;\;\;\;\;\;\;\;\text{Strain: }\frac{\Delta L}{L}}$ $\newline$ ##### *Beyond the elastic limit* - As long as the relationship between **displacement *(due to stretching)* and force is perfectly linear/proportional** an object will always **return back to its original position when the force subsides.** If the displacement gets **too large** this linear relationship will **change/deviate** which indicates that the object has become **permanently deformed**. *(Note that there is a small period of initial deviation, known as the elastic period where the object will still return back to it's original position when the displacing force subsides)*. - This *"maximum displacement"* is known as the **elastic limit** of an object. The **stress on the object at this point** is referred to as it's **tensile strength**. $\newline$ $\text{Tensile strength }=\frac{F_{max}}{A}\;\;\;\;\;\;\;\therefore\;F_{max}=(\text{\small tensile strength})A$ $\newline$ >[!bug] >##### *Note:* >- Its important to remember the **Hooke's law only applies when the relationship between the displacement and restoring force of an object is linear**. > - This can be confusing because objects *are able to stretch **past** this point and still return back to equilibrium*, not having undergone any plastic deformation. ##### *More on Young's modulus* - Youngs modulus is a value that **quantifies the tensile or compressive stiffness of a solid material when a force is applied lengthwise**. In other words, it is a quantity that tells you how hard a material is to deform. - ==A high Young's modulus indicates that the corresponding material is very **stiff**, meaning it cant deform much before breaking. A low Young's modulus indicates that the material is **elastic**, meaning it can deform quite a bit *before* breaking.== - Materials are called **rigid** if they experience only small changes in dimension under normal forces. - Materials are called **pliant** if the can be stretched easily or show large deformations when relatively small forces are applied. - **Young's modulus is measured in units of newton per square meter $\small N/m^2$.** This unit is also commonly referred to as **a pascal (pa)**. - If we know the ratio between the stress and strain on an object when it is being stretched we can find the value of **Young's modulus** for that specific material by using the equation below: $\newline$ $\text{Young's modulus formula: }\;\;Y=\frac{\text{stress }\sigma}{\text{strain }\epsilon }$ $\newline$ >[!quote] >##### *Graphing deformation vs force* >![[Springs and elasticitygraphingdeformation.png|850]] --- ## Comparing springs ##### *Springs in parallel* - Springs are considered to be "in parallel" when they are all connected to the same object and **all have the same length**. If the springs all have the same *spring constant* then they will all share *(or split up)* the load exerted on the object they are collectively connected to *equally.* - This means that the distance each, individual string stretches is reduced **in proportion to the number of springs that are in parallel**. - If there are two springs in parallel, each spring will experience **half the force** and therefore only stretch **half as far**. If there are *three springs* in parallel, each springs will experience a **third of the force and therefore only stretch a *third* of the distance** *(compared to just a single spring)*. - *Note: the above information assumes all the springs have the same spring constant.* - If there are multiple springs in parallel that all have the same length but each have a **different spring constant we cant use the method outlined above.** Instead, we must treat the system of parallel springs as a **single spring**. To find the spring constant of this *"combined"* spring all we have to do is add up the spring constants of each parallel spring in the system. Using this new "combined" spring constant we can preform our calculations as usual using *Hooke's law*. - *Note: this works even if the springs all have the same spring constant.* $\newline$ $\text{Spring constant of springs in parallel: }\;\color{LightCyan}\large k_{eq}=k_1+k_2+k_3\;...$ $\small\color{grey}\text{just add together each parallel springs' individual spring constant}$ ##### *Springs in series* - Springs are considered to be *"in series"* when multiple springs with **the same length** are connected **in line** with each other *(end to end)*. Unlike springs in parallel, **springs that are in series all experience the same load**, no matter how many springs you add. - If there are *two springs* in series, **each spring** will independently stretch by the same amount as a **single spring** experiencing the **same force**. This means that If there are *two springs* in parallel, their combined length will be **double** that of a single spring experiencing the **same force**. - If there were *three springs in parallel*, the combined length would be **triple** that of a single spring experiencing the same force. - *The above information assumes that all the springs have the same spring constant.* - To find the **overall spring constant** of a system that consists of multiple springs in series we can add together the **reciprocal of springs' individual spring constants**. Again this is useful for determining how much a system of springs in series stretches if the system is made up of multiple springs' **with different spring constants.** $\newline$ $\text{Spring constant of springs in series: }\;\large\color{LightCyan}\frac{1}{k_{eq}}=\frac{1}{k_1}+\frac{1}{k_2}+\frac{1}{k_3}\;...$ $\small\color{grey}\text{just add together the reciprical of each springs' individual spring constant}$ $\newline$ >[!quote] >##### *Springs in series and parallel* >![[springs in series and parallel.png|850]] --- ## Calculator <iframe src="https://app.calconic.com/api/embed/calculator/64889e9c28c7470029d3e4b6" sandbox="allow-same-origin allow-forms allow-scripts allow-top-navigation allow-popups-to-escape-sandbox allow-popups" title="Calconic_ Calculator" name="Calconic_ Calculator" height="5000" scrolling="no" style="width: 100%; border: 0; outline: none;"></iframe>