Conservative Force

A force is conservative when the work done on an object does not depend on the path it takes from its start to end position.

Concept bq

The work done by a conservative force depends only on displacement and is independent of its velocity or acceleration.

The work done by a conservative force on a closed loop is zero:
$\oint F\cdot dr=0$

Definition bq

The “conservative” potential energy of a particle/system is typically written using the potential function $V$.

---

Potential Energy

Potential energy is the stored energy of an object or system that represents the potential to do work. The work done by gravity and spring forces can be stored as potential energy ($V$).

The total potential energy is given by:
$V=V_g+V_e$

Potential Functions

A potential function (denoted with $V$) is a scalar function of position that represents the potential energy of a particle within a conservative force field.

Gravitational Potential Energy

Gravitational potential energy is given by:
$V_g=mgh$

Comment bq

$h$ is the height above any chosen datum.

Elastic Potential Energy

Elastic potential energy is given by:
$V_e=\frac{1}{2}k{s}^2$

Quote bq

$s$ is the deformation from the unstretched length.

---

Conservation of Energy

The conservation of energy states that energy cannot be created or destroyed. When a particle is acted upon only by conservative forces, the sum of kinetic energy and potential energy remain constant.
$T_1+V_1=T_2+V_2$

---

Force as a Gradient

A conservative force always drives an object towards regions of lower potential energy (downhill). The force vector is the negative gradient of a potential function.
$\vec{F}=-\nabla V=-\left(\hat{i}\frac{\partial }{\partial x}+\hat{j}\frac{\partial }{\partial y}+\hat{k}\frac{\partial }{\partial z}\right)V$

This means if the energy map ($V$) of a particle is known, you can calculate the exact direction and magnitude of force $\vec{F}$ at any coordinate.

Potential Energy Curves

By isolating velocity ($v$) from the conservation of energy equation ($T+V=E$), we can determine where a particle is physically allowed to move:
$\frac{1}{2}m{v}^2+V(x)=E⟹v=\pm \sqrt{\frac{2}{m}[E-V(x)]}$

When $E<V$, the speed is imaginary, thus indicating an impossible position for the particle.

Regions of a Potential Curve in a Well

A potential well is a region that confines a particle or object below a certain minimum potential energy (for example a half-pipe).

If an object does not have enough energy to escape the well, it oscillates between the two turning points $x_A$ and $x_B$ forever.
$x_A<x<x_B$
If an object does have enough energy to escape, it leaves the well and moves on freely past the barrier ($x_C$).
$x>x_C$

This means mathematically, $x$ can never be less than $x_A$ or between $x_B$ and $x_C$ as it must fall into either scenario.
$\text{Foridden}:x<x_A,x_B<x<x_C$

---

await dv.view("Scripts/topic-backlinks");