Why does a 741 op-amp get driven to the negative rail when the non-inverting terminal is left open?
The input bias current for a 741 is about 80 nA: every second, the chip is pulling a (tiny) bit of charge from that input pin. 

If there’s a resistive path, it can provide that current. 

But if the pin is open-circuited, it acts as a small capacitor. The removed charge drives the voltage at that point more and more negative. That drives the output negative. 

Eventually, all those voltages get negative enough that the op amp becomes internally unbalanced, and the output sits down near the rail.