--- ## Energy basics ##### *Energy transformations/transfers* - Energy *transformations* **are changes of energy with a system from one form to another**. - Energy *transfers* are different. They are the **exchange of energy between a system and its environment.** - **Work** describes the process of transferring energy to or from a system to an environment through **mechanical motion** *(kinetic energy)*. - **Heat** describes the process of transferring energy to or from a system via a **nonmechanical** means. - Since work allows us to change the energy of a system we can relate the total energy of a system to the amount of work done on it with the following proportional relationship. *This is known as the work energy equation.* $\newline$ $W=\Delta E=E_f-E_i\;\;\;\;\text{\color{grey}\small where}\;\;\;\;\Delta E=\sum\Delta E_{type}$ $\newline$ - *This equation illustrates why isolated systems have their energy conserved: without work, the total energy in a system cannot change.* ##### *Types of energy* - **Kinetic energy** is the energy of motion. All objects that are moving are said to have some kinetic energy. The **heavier an object is and the faster it moves, the more kinetic energy it is said to have.** - **Gravitational potential energy**, is *stored energy* that is associated with an object's **height above the ground.** The higher up an object is moved, the greater its gravitational potential energy. - When an object with *gravitational potential energy* is released it will convert that energy **into kinetic energy** as it accelerates towards the ground. - **Elastic potential energy** is the energy *stored* when a spring or other elastic object is **stretched out or compressed inwards past its equilibrium position.** - When an object with elastic potential energy is released the potential energy will transform into **kinetic energy** as the spring/elastic object works to move it back towards the equilibrium position. - **Thermal energy** is the sum of the **kinetic and potential energies** of all the molecules that make up an object. In other words, its a measure of **heat**, which is just itself, a type of energy. - **Chemical energy** is the energy stored by the strong and weak nuclear forces between atoms. This energy can be released when bonds between these atoms **are broken/re-arranged**. - **Nuclear energy** is the energy created due to the strong/weak nuclear forces that hold [[Electrons]] [[Protons]] and [[Neutrons]] together to form **atoms.** When these bonds are broken *(nuclear fission)* a huge amount of this *nuclear energy* is released. --- ## Kinetic energy - **Translational kinetic energy** is the energy associated with an object that is moving from one location to another. This kind of kinetic energy is denoted using the variable: $\small K$. - If an object is moving at **a constant velocity its kinetic energy will also be constant**. - Kinetic energy ==is always measured in units of joules $J$== - **Rotational kinetic energy** is the energy associated with an object that is **rotating** around an axis that lies **within itself**. This kind of kinetic energy is denoted using the variables and subscript: $K_{rot}$ - Both rotational and translational kinetic energy **increase exponentially** as the velocity *(or angular velocity)* of their associated object increases. $\text{Kinetic energy equations: }\;\;\Huge\boxed{\color{LightCyan}\large K=\frac{1}{2}mv^2\;\;\;\;\;\;K_{rot}=\frac{1}{2}I\omega^2}$ $\newline$ $\text{Change in kinetic energy: }\;\Delta K=\frac{1}{2}m\Big(v_{\small f}^2-v_{\small i}^2\Big)$ >[!tip] >##### *Interesting fact:* >- From these equations you should be able to see that the **overall kinetic energy of a *rolling object* will always be greater than that of a nonrotating object that is moving at the same speed**. --- ## Potential energy - Potential energy is the term we use to describe the **energy that is stored in an object due to its position relative to other object within a system**. We call this kind of energy potential energy because it has the ***potential* to be converted into other forms of energy** when the object it is associated with is released and allowed to return to its ideal state. - You can only gain potential energy through positive work. *Also* in order to store potential energy you need to continue exerting positive work on the system. ##### *Gravitational potential energy* - Gravitational potential energy is the energy that is associated with the height of an object. - ==Gravitational potential energy depends only on the height of an object and not on the path the object took to get to that position.== When the object is allowed to fall downwards, the potential energy is converted to kinetic energy. We denote gravitational potential energy using the variable: $\small U_g$ - Since it always takes some amount of force to change the gravitational potential energy of an object we can derive a very simple relationship between the gravitational potential energy of an object and the amount of work done on that object: - ==Potential energy is always measured in units of joules $J$== $\newline$ $\text{Change in gravitational PE: }\;\Huge\boxed{\large\color{LightCyan}\Delta U_g=mg\Delta y}$ $\color{darkgrey}-=================================-$ $\text{Relationship between gravitational PE and work: }\;(U_g)_f=(U_g)_i+W\;\;\;\text{-or-}\;\;\;\;\color{lightcyan}\Delta U_g=W$ $\text{}$ >[!quote] >##### *Gravitational potential energy* >![[gravitationalpotentialenergyvisual.png|850]] ##### *Elastic potential energy* - Energy can also be stored in a *compressed* or *extended* spring as **elastic potential energy** which is denoted using the variable: $\small U_s$ - Elastic potential energy is gained when an object is moved **against the restoring force of a spring or *other elastic object***. When this object is released the spring will exert a restoring force on the object converting the potential energy to kinetic energy. - The amount of elastic potential energy that is stored in a spring that is at some **stretched/compressed position is equal to the amount of work done to get the spring to that position**. - ==Potential energy is always measured in units of joules $J$== $\newline$ $\text{Change in elastic PE: }\;\;\Huge\boxed{\large\color{LightCyan}\Delta U_s=\frac{1}{2}k\cdot x^2}$ $\newline$ - Since it takes work to compress or stretch a spring we can also derive a very simple relationship between the change in the elastic potential energy of an object and the **work** done on that object: $\newline$ $\text{Relationship between elastic PE and work: }\;\Delta U_s=W$ --- ## Thermal energy - Thermal energy is the sum of the individual **kinetic and potential energies of all the atoms/molecules that make up an object**. Colloquially, we often call this quantity temperature, or heat. - *Note that thermal energy is note the **average** of the KE of the atoms that make up a substance, it is the **summation of their KE's***. - Friction between two objects **always increases their thermal energy**. If there is friction between an object and a surface the greater the **displacement of that object along that surface** the greater the amount of thermal energy that will be created. - Drag between an object and a medium also always increases the thermal energy of the object. - ==Thermal energy is always measured in units of joules $J$== $\newline$ $\text{Change in thermal energy due to friction: }\;\Huge\boxed{\color{LightCyan}\normalsize\Delta E_{th} = f_k\cdot\Delta x}$ $\text{Change in thermal energy due to drag: }\;\Huge\boxed{\color{LightCyan}\normalsize\Delta E_{th} = D\cdot\Delta x}$ $\newline$ --- ## Conservation of energy - Just like with [[Impulse|momentum]] energy is **a property that is conserved**. This means that the total energy within that system will **==never change, it will only ever transfer from one form to another== *(potential to thermal ...)*** Unlike with conservation of momentum **when using conservation of energy to perform calculations you can choose a non-isolated system** so long as you know the amount of work **done by or done on that system.** - If gravity is involved in any situation you must remember to always include **the earth in your system**. - To use the property of conservation of energy in our equations we just set the initial and final **energies equal to each other**. This allows us to solve for any unknown values. See the example below: $\newline$ $\text{Conservation of energy: }\;\color{LightCyan}E_i=E_f$ $\color{darkgrey}-===================================-$ $\text{Example conservation of energy equation: }$ $K_f+U_f+\Delta E_{th}=K_i+U_i+W=\Huge\boxed{\color{LightCyan}\normalsize \frac{1}{2}mv_f^2+mgy_f+\Delta E_{th}=\frac{1}{2}mv_i^2+mgy_i+W}$ $\newline$ - The **total energy** of a system is the sum of all the different types of energies within that system. If the system is isolated *(meaning that it does not interact with any outside objects/forces)* this value should **never change**. ==Therefore, the total energy of an isolated system is always conserved==. - The heat transfer $\small Q$ and the work done $\small W$ must be equal to zero for a system to be **truly isolated**. - Since energy can only transfer **in/out of a system through work or heat transfer we can create the following equation** that relates the change in energy *(of a system)* and the work/heat transfer done on a system. - This equation shows that in isolated systems *(that have 0 work done on them and 0 heat transfer)* also have 0 change in energy. $\text{Change of energy equation: }\;\Huge\boxed{\color{LightCyan}\large\Delta E=W+Q}$ $\newline$ >[!quote] >##### *Conservation of energy visual* >![[conservation of energy.png|850]] --- ## Calculator <iframe src="https://app.calconic.com/api/embed/calculator/648cfa1928c7470029d3f4b2" 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="10000" scrolling="no" style="width: 100%; border: 0; outline: none;"></iframe>