>[!summary] Energy is always a conserved quantity > Potential energy is the amount of energy in a system > Energy transfer is the transfer of energy to different systems > >Power is the rate at which energy is transfered over time > **Key equations:** > If force is constant $W = F\Delta \cdot x$ > If force is non-constant: $W = \int_r ^{r_f} F(r) \cdot dx$ > Work-energy theorem: $W = \Delta K$ > Power if its constant: >$P = \frac{F(t) \cdot dr}{dt} = F(t) \cdot v(t)$ > >$P_{avg} = \frac{W}{\Delta t}$ > Power if its non-constant: $P_{avg} = \frac{1}{\Delta t} \int_t ^ T P(t)dt$ # Potential Energy **Energy is always a conversed quantity**. It may not always be constant. In a closed system there is no energy transferred out or in to allow it to be conserved. Potential energy is the amount stored in a system. >[!info] Potential Energy In general system always want to have the lowest amount of energy. Since its energy favourable. In [[Electric Potential]] the most favourable position an electron can be in is when the the electron is near an opposite charge object. In [[Gravitational Potential energy]] the most favourable position an object can be in is when a object is in a state of equilibrium. # Work ![[wor_1.png]] [^1] >[!note] Explanation 1D case of energy transfer over a distance In a 1D case energy transfer is the amount of effort to transfer done on a object (work). If we do work over a distance x than we donate the amount of energy transfer by (In 3D he force and distance are dot products) $W = F\Delta x$ >[!warning] Assumption If we assume F is constant over a distance If F is constant we can use [[Kinematics]] so that $ax = \frac{1}{2}(v^2 - v_0 ^2)$ so that: $\begin{array}{c} W = F\Delta x \\ W = ma\Delta x \\ W = \frac{1}{2}m (v^2 - v_0 ^2) \\ W = \Delta K \end{array}$ # Non-Constant Force (Work-energy) For when force is non-constant we cant assume that force is constant. So we need to integrate over the whole distance. ![[wor_2.png]] [^1] >[!note] Explanation >Example of Non-constant force $\begin{array}{c} W = \int_r ^{r_f} F(r) \cdot dx \\ F(r) = ma = m\frac{dv}{dt} = m \frac{dv}{dt} \cdot \frac{dt}{dx} = m v\frac{dv}{dx} \\ W = \int_r ^{r_f} mv \frac{dv}{dx} dx \\ W = mv \int_r ^{r_f} dv \\ W = \frac{1}{2} m(v_f ^2 - v^2) \end{array}$ **Note that you get the same solution if the force is constant.** # Power Power is the rate at which energy is transferred over time, with a force and distance being applied. We denote this by (if force is constant): $P = \frac{F(t) \cdot dr}{dt} = F(t) \cdot v(t)$ ![[wor_3.png]] [^1] >[!note] Explanation If force is not constant in power example If force is not constant like before we need to integrate: $\begin{array}{c} P_{avg} = \frac{1}{\Delta t} \int_t ^ T P(t)dt \\ P_{avg} = \frac{1}{\Delta t} \int_t ^ T \frac{F(t) \cdot dr}{dt} dt \\ P_{avg} = \frac{1}{\Delta t} \int_t ^ T F\cdot dr \\ P_{avg} = \frac{W}{\Delta t} \end{array}$ If we assumed force was constant we would find the same result [^1]: Taken from R. Epp Lecture notes.