# Tension at lowest point: oberbeck pendulum

I'm solving a problem involving oberbeck pendulum

We are given $$M$$, $$m$$, $$r$$ (radius of pulley), $$R$$ (distance from center of rotation to each of the 4 small weights). The problem says that the weight descends from a height $$h$$, reaches the lowest point and then begins to rise.

We are asked to find the tension $$T$$ during the time that the weight is changing directions.

This are the equations of motion for the system: $$I=4mR^2$$ $$I\frac{dw}{dt}=Tr$$ $$Ma=Mg-T$$ $$\frac{dw}{dt}=\frac{a}{r}$$

I managed to find acceleration $$a_0$$ and tension $$T_0$$ during the time that the weight is descending/ascending, but I I'm stuck at finding the tension during the time that it is changing directions. $$a_0=\frac{Mg}{\frac{I}{r^2}+M}$$ $$T_0=\frac{Mg}{1+\frac{r^2M}{I}}$$

I found a solution that says this: during the time of the "yank", the pulley makes a $$180^\circ$$ turn, not changing its angular velocity (given the high moment of intertia). So we get: $$(T-Mg)\Delta t=2Mv$$ $$\Delta t = \frac{\pi r}{v}$$, and $$v$$ is the velocity when it reaches the lowest point.

As I understand, the pendulum keeps rotating, since the torque created by this change in tension $$\Delta T$$ is small, taking into account the moment of intertia of the system. But why is it that the tension $$T=T_0+\Delta T$$ acts only during this $$180^\circ$$ rotationi and then it goes back to being $$T_0$$? It kind of makes sense at an intuitive level, but I don't know what physics law backs it up.

It's my first time posting here, thank you in advance for your help!