First side room with gears and weights/ Climbing to the upper ledge
Approach either weight. When it comes down, jump to grab the first handhold of the weight, then the next one.
When it reaches the top, jump back to grab one of the moving horizontal poles that connect the highest gears.
Wait until it reaches the highest point. When it does so, pull up on the pole and jump up to grab the chain that moves towards the ledge you are headed. Once you reach it, drop onto it.