Poential Energy is represented by the symbol U.

U = mgh

W_{by Gravity}= -ΔU

ΔU = U_{Final}– U_{Initial}= 0 – mgh = -mgh

ΔU = -mgh = W_{by Gravity}= ΔKE

What is important is the change in potential energy.

-ΔU = (U

_{Final}– U_{Initial}) = W_{done by Gravity}= ΔKE = KE_{Final}– KE_{Initial}

_{ }

Lost Potential Energy = gained Kinetic Energy

KE

_{Final}+ U_{Final}= KE_{initial}+ U_{Initial}

Total Mechanical Energy is Conserved

Conservative Force Field: Total work done about a closed path sums to ZERO.

Friction Force is non-conservative (You do not get energy back in a closed path)

Imagine how force looks as a ball is dropped from the sky. think of *dy* as the many very small distances it travels over to fall to the ground. dW is the very small but of work

ΔU = -W = -F

dU = -dW = -F_{y }dydU/dy = -F_{y U(y)=mgy }

Read this “The potential Energy with respect to height is equal to mass times gravity times height.

Now what about stretched springs? This story is a little different. The force exerted to do work grows as the spring stretch’s further.

If you draw a graph, the potential energy of the spring as it is compressed and stretched would form a parabola.

Now what about a child on a slide?

The component of gravity parallel to the slide is mg cos θ. It is doing work. θ is changing all the time, as the mass of the child moves along the slide.