Potential Energy - Yousef's Notes

Conservative Forces We hence define the potential energy in each case as

Gravitational potential energy	Elastic potential energy
$U_{\text {grav }}=m g y$	$U_{\mathrm{el}}=\frac{1}{2} k x^{2}$
$W_{\text {grav }}=U_{\text {grav, } 1}-U_{\text {grav, } 2}=-\Delta U_{\text {grav }}$	$W_{\mathrm{el}}=U_{\mathrm{el}, 1}-U_{\mathrm{el}, 2}=-\Delta U_{\mathrm{el}}$
Potential energy increases as the body moves up	Potential energy increases as the spring is stretched or compressed
![[Pasted image 20240306140057.png]]	![[Pasted image 20240306140105.png]]

Remarks:

Potential energy can be interpreted as a measure of the possibility for work to be done.
Same units as work and kinetic energy.
Gravitational potential energy is a shared property between the body and the earth

#Gradient force

A more systematic/mathematical way to define potential energy is when a force can be written as the gradient of another scalar function

$$ \vec F=-\vec{\nabla} U=-\left(\frac{\partial U}{\partial x} \hat{\imath}+\frac{\partial U}{\partial y} \hat{\jmath}+\frac{\partial U}{\partial z} \hat{k}\right) $$ Then $$ W=\int \vec F\cdot d\vec r=\int -\vec{\nabla} U\cdot d\vec r=-\int dU=U_1-U_2 $$

Hence we conclude

any minimum in a potential-energy curve is a stable equilibrium position. Conversely, any maximum in a potential-energy curve is an unstable equilibrium position.
a conservative force always acts to push the system toward lower potential energy.

#Energy Diagrams

![[Pasted image 20240306140151.png]]