Can I operate 60Hz (3600rpm) generator at 50Hz (3000rpm)?

If you operate a 3600 rpm (60 Hz) synchronous generator at 3000 rpm (50 Hz), the power output will be reduced, simply because the mass of the rotor remains constant and you reduce its rotational speed. Based on the laws of physics, kinetic energy is directly proportional to the mass and the square of the velocity. Conversely, if you operate a 3000 rpm (50 Hz) generator at 3600 rpm (60 Hz) the power output will increase, PROVIDED, the rotor and the stator can withstand the increased electrical and mechanical stresses due to the increased power output and rotational speed. Regarding the efficiency, it will depend on the design of its electrical winding, stator core, air gap, and mechanical bearing components. It is also ASSUMED that your prime mover (steam turbine) can also withstand the increased mechanical stresses due to the increased rotational speed.

Note that 3000 RPM is the 50 Hz equivalents of 3600 RPM at 60 Hz. Ultimately this is an economical design decision. Remember the number of poles is reflective of the amount of copper utilized in the construction and the speed is a function of the amount of fuel consumed (consider also optimum conditions for the operation of the combustion process) as well as the lifetime expectations of the system components (think bearing life etc.)

There is a big believe that the frequency Hz has to be 50Hz (or 60Hz), but this is not true. If you have standalone generators driving an offshore platform, there is no need to have 50,000 Hz, but to have about 50Hz average over time. There is absolutely no equipment on an oil platform (or any industry) plant that can not function properly in the region of 48-52 Hz, as long as you have your average at about 50Hz. I have seen numerous failures related to aggressive settings on generator load sharing units to strive to maintain 50Hz at all costs, resulting in unstability of the whole plant when several generators have to be able to collaborate to supply the load.

On shore when connected to the main grid, there is another issue as the grid is so large compared to your generation that the frequency can be considered constant (changes in 0,0001 Hz range in good places). Here you don't need to be so aggressive as the net will not allow you to change the net frequency.

If your system frequency drops from the rated frequency, whether 50 Hz or 60 Hz, it indicates that your real power (kW) demand exceeds your real power (kW) generation. You can normalize the frequency by increasing the power input of your prime mover. Operating any electrical or electronic equipment far below or above its rated frequency will affect its performance or worse, it can be damaged.