My lecturer gave us some exercises on Dynamics with explained solutions in case we get stuck. One of the exercises is this:
A particle of mass $\space m \space$ is attached to the end of a light inelastic string of length $\space a \space$, the other end being fixed.
Initially, the particle hangs freely vertically below the fixed end, and is given a horizontal velocity $\space u \space$. After the string has turned through an angle $\space \theta \space$, show that the tension in the string is:
$$ m \left[g \left (3 \cos{\theta} - 2 \right) + {{u^2}\over a} \right]$$
My lecturer handwrites the solutions. Here is a snippet of what he wrote and what confuses me:
Notice how he subbed the $\space \ddot{r} \space$ with velocity $\space u \space$, but also subbed $\space {d \dot{\theta} \over dt} \space$ with velocity $\space u \space$.
Also, I can't tell whether he wrote $\space u \space$ or $\space \dot{u} \space$. If it's the latter, isn't $\space u \space$ just a constant and not a velocity function of time? Hence making the act of dotting it pointless?
Sorry for the overload of questions, I'm really stuck. I much rather prefer typed notes to avoid issues like this.
As I pointed out in another answer, your lecturer seems to be writing "i.e."
You claimed that in the second of your confusions that he replaced $\frac{d\dot\theta}{dt}$ with $u$. This is not what happens in that step. Here is what happens: $$\require{cancel}\begin{align}\frac {\cancel{m}}{r}\frac d{dt} (r^2\dot\theta)\hat\theta&=\cancel mg\sin(\theta)\hat\theta\\\frac 1r(2r\dot r\dot\theta+r^2\ddot\theta)&=g\sin\theta\\2\dot r\dot \theta+r\ddot\theta&=g\sin\theta\\\end{align}$$ Then if you use the fact that $r=a$ does not chance then $\dot r=0$, and this gives $$a\ddot\theta=g\sin\theta$$ which is what your lecturer looks to have written.
Also as I wrote in a comment of that answer, later in the text, the lecturer uses 9.15 in 9.13, and there is no trace of $u$ or $\dot u$. Finally, looking at the way your lecturer wrote $u$ in the last line suggests that he does not write it with a curl at the end, as all the circled ones have. So the circled ones appear to say i.e.