Im powering steppers and UNO with a 12V/5A adapter.
The adapter doesn't have a ground plug.
The components are enclosed in a metal case, does it need to be grounded, if so how?
It's not necessary but you can connect the case to the negative side of the power supply which is he Arduino's ground.
Obviously that's not earth ground, unless you connect USB and there is a round-about earth-ground through your computer. (Most laptops don't have an earth-ground but most desktop computers do.)
A local-ground to the case might help to shield electro-magnetic noise from the motors from "leaking out" into nearby radios. (Stepper motors and motor drivers are electrically noise, as well as somewhat mechanically noisy. )
It's more common to have a shielded box, or a shielded cable, when you have something like a sensitive analog audio circuit and you want to keep noise out. In this case a local ground is just as effective as earth-ground, and in fact an earth-ground sometimes introduces a ground-loop (multiple ground paths) and that can introduce noise into your sensitive circuit.
...In general, digital electronics are highly-immune to noise.
A metal box is normally earth-grounded as a safety-ground when AC power goes into the box. This is just a safety precaution but it's sometimes required on commercial products. If an AC wire comes loose and touches the box, the voltage is shorted to ground which means the case can never be "live", a breaker blows, and it's safe. Without the ground, the box can become energized and of course that's dangerous (with lethal AC voltages).
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No, the ps provides galvanic isolation.
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It does, but might not if there's a fault.
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Right, but (at least where I live and to my understanding of the regulations on fault protection) that doesn't mean you have to ground a metal case just in case of a PSU failure.
For example, a model railway has a bunch of bare metal which is "only" protected from dangerous voltages by the galvanic isolation in the power adapter. They are still allowed to be sold.
The same goes for your computer - there's a bunch of exposed contacts on each of your USB ports, none of them are protected against accidental touches, they are not grounded (and if they were, they wouldn't work) and they are "only" protected from high voltages by the galvanic isolation in the power supply unit.
And, of course, the Arduino itself has a bunch of exposed contacts - if it was plugged into a faulty PSU via barrel adapter, the soldering pads on the voltage regulator could give you a fatal zap.
But all of the devices I named above are safe and conform to the relevant norms because it is generally safe to assume that a properly-constructed AC/DC power adapter will only provide its specified output voltage, and the risk of its galvanic isolation failing is laughably low.
So, as long as the PSU is outside of the metal case, and everything within the metal case will only see DC voltages that are safe to touch, I don't think it needs to be grounded.
(Note: People often treat AC as inherently unsafe and DC as inherently safe - this is wrong, DC can also hurt or kill people at sufficiently high voltages. Only low-voltage DC is safe to touch - but 12V is absolutely fine.)
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Failure is all but eliminated if you use a UL approved PS. I use these MEAN WELL GSM18U05-P1J ones that are "medical grade" (whatever that means) and have world wide approvals. 5V @ 3A
Don't bet on it. 
If they have a full "clover" plug on the brick rather than just a "figure eight", then they will have a protective ground.
Uhh, no. The ground plug is not ohmically connected to the output. This is what we mean about "galvanic isolation". Ohm for yourself. There shouldn't even be hipot leakage.
In that case, the brick has a protective ground - the laptop itself hasn't. Big difference.
No, it could be either way, you can learn nothing without knowing the inside of the brick.
If the notebook has a two-pole barrel plug (as most do), I think it's a reasonably safe bet that it doesn't have a protective ground even without taking apart the PSU.
There is absolutely no justification in electronic terms for saying this.
Yes there is. UL, CE, etc all require ISOLATION between the line and load. Ohm to your hearts content, you'll NOT find a PS with a barrel connector that has any path to ANY input pin. It's a safety thing. Now with a PC that has a METAL CHASSIS with LINE VOLTAGES inside it, yes the CHASSIS must be grounded. But with wall warts there is ABSOLUTELY ZERO input to output coupling. This is what "medical grade" means, there are ZERO leakage paths that could EVER POSSIBLY connect line current to the output and cause cardiac arrest (very low mV's and mA's). This is a safety reg. and EU is tougher than US so "worldwide" safety approvals means just that. Intrinsically safe even during failure.
I am so baffled by this that I don't even know how to argue against it.
You cannot fit a protective ground onto a two-pole barrel connector or any other two-pole connector if that connector also has to fit a regular ground and a live wire. If you disagree, please explain how you would achieve this technological breakthrough.
Three is a bigger number than two.
Protective ground / protective earth is also not the same thing as your regular ground, and if you connect the two, you do not have a protective ground.
Your bafflement is due to miss reading what I put.
If you follow the thread Paul was suggesting.
You then disagreed with that, on the spurious grounds of having a two pole switch on the brick to laptop so I responded saying that you can't tell if the protective grounding was connected to a two pole switch without knowing the circuit inside the block.
So note I am NOT talking about a figure of 8 plug.
I decided not to pursue this matter further, as I picked up the nearest "brick" with a "clover" plug and was unable to find continuity between the "earth" pin and the output negative.
I most certainly have had problems in the past with audio recording and "ground loops" through the "earth" pin of a power "brick". These are USFF PCs, not necessarily laptops as such so one wonders why they have an "earth" connection if they have no metal case and are effectively, double insulated.
Maybe the original spec was for a metal case, so the electronic designers worked the earth safety ground into the design. Then the spec was changed for a plastic case and the ground was not reconsidered or was deemed not worth changing.
No, I didn't. I didn't mention two-pole switches even once. And nobody else did, either. It's pretty audacious of you to accuse me of "miss reading" you while you are just making stuff up out of thin air, and honestly I think it's a bit offensive.
My point here was that just because the external PSU (power brick) has an INPUT with three poles (e. g. "cloverleaf") and a protective earth does not mean the laptop has a protective earth, and if the line connecting the PSU to the laptop (which is, in my experience, almost always a two-pole barrel plug regardless of the connection used on the PSU's input side) is only two-pole, then the laptop itself cannot have a protective earth.
Exactly the sort of mistaken understanding I said you have.
What if the three pin plug of the power block has a protective ground and that was connected inside to the negative rail of the two pin plug. Then the laptop would have a protective ground.
What I am saying is that you can't assume that has not happened without a schematic of your power block. The fact that you have one or even 50 that has not got this, it doesn't mean that you can't get them wired like this.
Sorry that was a typo it should have read two pin plug.
No, it wouldn't. It would have a regular ground which is wired wrong. And this would just flip your ground fault protection (if you have one) so your power block just wouldn't work.
You have a point in that some devices may break regulations so you can't just assume a device is compliant without taking it apart. But, by the same token, you can't tell if your PSU contains a hidden bomb without taking it apart, so if you want to be 100% sure, your options are to a) buy from trustworthy manufacturers and assume they know what they're doing, or b) live in a cave without anything electronic.