How Do Kinetic and Potential Energy Differ in Physics?

[girlwhoneedsmathhelp]

Homework Statement

I saw this in my school textbook and have no idea how they manage to get this equation :

The context is Circular Motion :
"A particle on a string, w(angular velocity) varies throughout the motion. As you saw earlier, the value of w at any instant is given by the energy equation, which in this case is :
(1/2)m(r^2)(w^2) + mgr(1+cos(theta)) = (1/2)m(u^2) where u is the speed of the particle at the lower point."

Thank You!

Relevant Equations

1/2mv^2
mrw^2

I know that (1/2)m(u^2) is KE and initially I thought this showed PE=KE but I don't think so anymore...
I believe this has something to do with acceleration and Centripetal force but I'm so so confused

 

[archaic]

I can make sense of this if the configuration is like this ($+$ and $-$ represent the sign of $\cos\theta$):

 

[haruspex]

I thought this showed PE=KE

No, it shows PE+KE=constant.
It might be clearer if we use the subscript 0 for values at the lowest point and v for the speed at any instant. So their u is my v₀.
Then we have rω=v, at all times, and PE+KE=$\frac 12mr^2\omega^2 + mgr(1+\cos(\theta))=\frac 12mv^2 + mgr(1+\cos(\theta))$.
At the lowest point, $\theta=\pi$, the PE is zero (by choice) and the KE is $\frac 12mv_0^2$, so $\frac 12mr^2\omega^2 + mgr(1+\cos(\theta))=\frac 12mv_0^2$.

 

[girlwhoneedsmathhelp]

Ah yes I see thank you both!