The answer to your question is not so much for a coffee forum but a very simple electrical problem that afflicts literally 95% of all single phase motors eventually, particularly motors that have higher torque duty like pumps.
Electric motors operate by creating offset magnetic poles that cause the rotor to spin. Starting this spin process is by far the most difficult task for an electric motor, like the effort to crank something from standstill.
This is far easier in 3 phase motors because you have different simultaneous vectors at your disposal. A reason 3 phase motors are more efficient in both startup and eventual lifespan. They can also be built a bit more compact.
Single phase motors will generally have one or two capacitors. The first capacitor is used for startup and if it's got a second, smaller capacitor it's function is for power factor correction (which is useful if there are a lot of motors hooked up).
The startup cap creates a phase split to enable the windings of of motor to think it's a 2 phase supply for a very short moment, or delay. Once the motor is spinning, the startup cap is essentially redundant as there is usually a centrifigual switch that controls the on and off for the start cap......sometimes you may hear a faint click sound as the motor is slowing after turning off a motor, that's the cfg switch reengaging.
Generally speaking, when motors fail with a pop, it's likely the cap has blown......for a variety of reasons, but caps don't last forever. Another cause of a blown cap is the cfg switch stuck on, meaning the start cap is always active.
Finding a replacement that has similar ratings is usually simple. Sometimes this caps are on the outside of the motor and in larger motors they are inside the terminal housing.
There are instances whereby a start cap can partially fail, or render itself less effective and motors continue to start, but the motor itself can sustain damage as the additional current draw risks burning out windings. The instantaneous load to spin a motor from standstill is a quantum larger than the load at run speed. If a cap is not working then this start load melts winding very quickly and can take out other components in the circuit as well, like expensive control boards.
Motors with light loads, e.g. Fans may be able to start.....sometimes they do this slowly spinning until rpm is sufficient to get enough polar momentum. Many bathroom and toilet exhaust fans suffer this problem prior to total failure.
Motors with higher loads like short burst pumps should not be used without the start cap as it will lead to motor burnout.