02 - Notable Potential Energy Calculations

With the last section's definitions, we can apply this concept to a few simpler, basic cases which act as fundamental building blocks to the efficient and simplified solution to many more intriquite problems we may face in the future.

Elastic Force Work:

if a spring exerts a force proportional to its deformation while in its elastic phase (the one which we will consider is the case for all relevant problems), we may say that:

$$ F_{elastic} = F_{el} = -k\Delta x = -kx$$

by convention, we say that the distance has a sign opposite to the force applied in the situation. This is convention, and should be studied carefully in every problem. This convention is helpful because it makes it so that the sign of the force and the direction it tends to is the same usually, rather than opposite [poorly worded].

thus:

$$ W_{F_{el}} = \int^b_a{kx \cdot dx} = \frac{kx^2}{2} \Big |^b_a $$

note:

$$ W_{F_{el}} = \frac{kx^2}{2} \Big |^b_a = (\Delta E_{P})_{el}$$

Gravitational Work:

Similarly, we can find the gravitational force work and the gravitational potential energy:

$$ \vec F_g = G \frac{Mm}{\vec d^2} $$ $$ W_{F_g} = \int_{a}^{b} \vec F_g d\vec r $$ $$ W_{F_g} = GmM \int_{a}^{b} \frac{1}{\vec r^2} d\vec r $$ $$ W_{F_g} = \frac{GmM}{r} \Big |^b_a $$ $$ W_{F_g} = (\Delta E_{P})_{g} $$

On earth for sufficently low altitudes:

$$ (\Delta E_{P})_{g} = mgh$$