<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/724202a9-05cb-4ba9-96f8-0c0ba9b496ce/Potential_Energy.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/724202a9-05cb-4ba9-96f8-0c0ba9b496ce/Potential_Energy.png" width="40px" /> Definition: Potential Energy is the energy stored by an object because of its position relative to other 🌏 objects, 🍾 stresses within itself, ⚡ its electrical charge, or other factors. For gravity:

$$ \text{General: } U_\text{grav}=-\frac{GmM_E}{r} \quad\text{For: }R\approx R_{earth}\;\rightarrow\; U_{grav}=mgh $$

</aside>

<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/5fea9ed9-4148-4a2e-a2b7-b2ee1ecd5e08/Notation.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/5fea9ed9-4148-4a2e-a2b7-b2ee1ecd5e08/Notation.png" width="40px" /> Notation:

$$ U=E_p \quad;\quad K=E_k $$

</aside>

Elastic Potential Energy:

$$ W_{tot}=W_\text{grav}+W_\text{el}+W_\text{all other}=\Delta K=K_\text{2}-K_\text{1} $$

Using this we can derive:

$$ W_\text{other}=(K_2+U_{2})-(K_1+U_{1}) \quad \text{where:} \\ U_1=\text{initial }E_p \; ; \; U_2=\text{final }E_p \; ; \; K_1=\text{initial }E_k \; ; \; K_2=\text{final }E_k $$

Definition: The work done by all forces other than the gravitational and elastic forces equals the change in the total mechanical energy of the system.

Conservative forces

<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/7735cf07-6323-4721-a80c-43337d1e6cfa/Consergvative_force.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/7735cf07-6323-4721-a80c-43337d1e6cfa/Consergvative_force.png" width="40px" /> Definition: A conservative force is a force where the total mechanical energy of the system is conserved Example: gravity ✅ ; elastic force ✅ ; friction 🛑✋ Properties:

• The work done by a conservaitve force is reversible
• Work done on moving round a closed path under influence of a conservative force is zero
• The work done by a conservative force is independent of the path, and depends only on the starting and finishing points
• The work done can always be expressed as the difference between the initial and final values of the potential energy function $U(x)$ </aside>

From this we can deduce:

$$ W_\text{tot}=W_\text{cons}+W_\text{non-cons}=\Delta K $$

Potential energy function for 1 dimension 🕹️

For conservative forces: $U(x)$ potential energy depends only on position

$$ \Delta U=-\int^{x_2}_{x_1}F(x)\,dx $$

if we take $x_1=0$ then we can find the initial formulas:

• gravitational force: $U(z)=mgz$
• elastic force: $U(x)=\frac{1}{2}kx^2$

In this case we can do it the other way around and deduce:

$$ F(x)=-\frac{\text{d}U}{\text{d}x} \\ \left . \begin{array}{ll} \bullet \;\; \text{if }F(x)>0\rightarrow U(x) \downarrow \\ \bullet \;\; \text{if }F(x)<0\rightarrow U(x) \uparrow \end{array} \right \}\text{ conservative force push system to low potential energy} $$