so i just got this problem earlier when im trying to fix my Laptop fan because it creates a loud noise everytime i turn on my laptop so i decided to unplug it in my laptop and leave the other fan working, but after i did that my laptop randomly shutting off. http://prntscr.com/nizsx0
plug it back in, your laptop most likely is overheating
the noise at powering on is totally normal, it´s because the system checks the fan speed while posting or while powering on the energy saving options are not loaded yet or it tries to cool the heat spike at boot up.
Your laptop gets really hot in normal conditions with the fan, without this, your laptop can´t hold on to lower temperatures.
If the fan is still unplugged and you want it like this, you will have to undervolt the CPU if possible. The power plan should be set to energy saving or similar and not high performance