Okay, grab a cold one as I'm going to be long winded here.
The starter motor is only one item of that system. You have a keyed or push button switch, which controls a start relay (start solenoid) which, in turn, routes power from the ships battery to the starter motor. And all these items are connected with wires, cables, and the airframe itself. AND, each wire or cable has terminal ends crimped on each end. All this means that there are a bunch of spots that need to be trouble shot if the system is inoperative.
As MCS Mike pointed out above, a quick way to trobleshoot is to just bypass the bad spots. An example is, if the starter motor turns slow with the key, but fast when jumping the battery positive directly to the main starter terminal, then there is a 'bad connection' on the positive side of the system. If you jump the battery ground directly to the starter motor case, then the problem is on the negative (ground) side of the system.
The more technically correct explanation of these problems is that a poor connection or bad switch/relay will cause a voltage drop. A voltage drop is just a point in a circuit that has a higher resistance than it's supposed to have. As the formula for voltage is amps X ohms, the current flowing across this resistance multiplied by the resistance, will give a voltage. THIS number is the voltage dropped across the bad component or connection. If we have a 12 volt battery installed, subtract this voltage drop from 12 volts and that is what will be left over to power the starter motor. This voltage drop can also be measured with a digital volt meter, with one meter lead placed on each side of the resistance. Example of this: A start solenoid has .1 ohms of resistance across the main contacts. If the starter motor draws 80 amps of current when cranking, the voltage drop will be 80 X .1 = 8. That means that after 8 volts drop at the start relay, only 4 volts are left to power the starter motor. So obviously the motor wouldn't even turn over in this case. Also notice that when drawing high current, the voltage drop is magnified. I.e., if we only drew 3 amps across the same resistance of .1 ohms, now the voltage drop would be .3 volts. Remember this, as, if the poor connection is at the battery connections, the voltage drop may not be noticeable under normal operations but still prevent proper operation of the starter motor.
A fairly quick way of checking for voltage drops is to place the negative lead of the meter directly on the negative post of the battery. Touch the positive lead to each cable connection, starting from the battery positive and working toward the starter motor. Note the voltage at each point. You are looking for battery voltage at each measurement. If you see a drop of more than a tenth or two of a volt, note that location. There is a high resistance between this point and the last point that had good battery voltage. It may be a poor connection, poor crimp of a cable terminal, or even a worn out relay. If you suspect an item like the start relay, you can verify it by placing one meter lead on each main POST (NOT the ring terminal attached to it). And remember that the system must be energized to check voltage drops. In other words you need to be cranking the engine.
Checking for poor ground connections is done the similarly. Place the positive meter lead on the positive battery post. While cranking the engine place the negative meter lead on the crank case. If you have a voltage reading here, there is a poor ground connection between the engine and airframe. Connecting the battery negative directly to the crank case is a good way of minimizing connections between the starter motor and the battery negative. If you have the battery negative connected to the airframe you'll need to remember that there may be several bolted or riveted connections in the airframe, that the ground path will need to cross.
Other wiring such as the small control wires from the breaker, to the start switch, to the start solenoid, can also cause problems. A worn out switch or corroded connection can mean no or intermittent operation. Those are usually easy to verify by using a voltmeter but remember that the circuit needs to be energized when doing voltage checks. In the case of checking the small control wires, it's usually safest to disconnect the large cable between the start solenoid and the starter motor. That way the start switch can be pushed to energize the control circuit without cranking the engine.
Don't forget the battery! If you measure a good 12 volts, at rest, and it drops to something like 9 volts when cranking, the battery may have a high internal resistance. If it doesn't improve with a top charge, swap it out with a known good battery. As batteries age the cells and chemicals lose there effectiveness and the battery will not provide rated output.
All this adds up to not delivering the correct power to the starter motor. And if this happens repeatedly it can cause the starter motor to overheat. Even the bad connections can get hot when energized which makes the resistance even greater.
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