What makes the potentiometer not a good component to use for reducing the 9V to 5V DC for an Arduino?
I thought of using it but I been told it is not a good idea, so I am looking into the reasons why.
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When I was young and innocent I tried the same for a bunch of relays, and my potentiometer caught fire!
First off, as you allude to, it is in some cases possible to regulate load voltage with a potentiometer. You can use it as a divider (three terminals) or as a series resistor (two terminals). However, this involves you adjusting the potentiometer depending on the drawn current. For constant loads this is feasible, but not so for fluctuating loads from a processors, as others here pointed out.
Your circuitry, including it's input/outputs and peripherals, use different currents depending on their state/mode. And their reliable operation depends on a constant load-independent supply voltage. That said, if the current fluctuation and the ensuing voltage fluctuation do not affect the operation of a (perhaps fixed) load, then the potentiometer or voltage divider can be used.
But there are other factors too: heat and efficiency. The current through the potentiometer, and the ensuing I^R heat, generates heat in both configurations. And in the divider configuration you have a current running even if there is no load.
Since my pyrotechnic experience, I instead use a "LM7805" integrated regulator for simple cases (https://en.wikipedia.org/wiki/Voltage_regulator)
These regulators keep the 5V constant practically for all (fluctuating) current loads from your Arduino and any peripherals (sensors, LEDs etc..). They are very easy to use, and quite suitable for a simple table-top home lab, but they are not very efficient. In other applications they can get hot, and so now I only occasionally lightly burn a finger.
If power efficiency / battery life matters, look at switch mode power supplies, like the LM2674
The potentiometer is a resistor component. When you try to create a voltage reduction through a resistor the voltage drop is 100% dependent upon the amount of current flow through the resistor. A load like an Arduino is not a fixed current load so it is not possible to use a single resistor value to reduce the 9V to 5V suitable for proper operation of an Arduino.
You will need to investigate what a voltage regulator does and why it is applicable to the application.
Beware that if your 9V source is a typical 9V battery it is far from a suitable long term power source for an Arduino due to the relatively low capacity of such a battery.
You may use as potentiomenter to provide an analog input signal to an Arduino (preferably with an additional resistor in series to prevent pulling the input above 5 volts), but you should not use a pot to provide power to an Arduino, for the reasons given in other answers.