The ground prong on my backpack vacuum came off and stuck inside a wall outlet. Should I replace the plug or does the ground not matter much? The plug itself gets really hot now when I use the vacuum.
Thanks for advice.
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The ground prong is important. If a failure occurs that connects the hot to the parts of the vacuum that are supposed to be grounded, a functional ground prong means the circuit breaker trips. Lack of it means you get to play with electrocution. Devices that do not have a ground prong from the factory are "double-insulated" meaning that two separate failures would have to occur to connect the hot to the case (and in most actual product, have non-conductive plastic casings so you cannot touch any metal parts)
The plug overheating is an additional reason to replace the plug and would be even if the ground prong was intact.
So, replace the plug.
Yes to the plug, but you should also change the outlet if you can't get that prong out.
It depends. many vacuum cleaners have no ground conductor in the power lead and the ground pin on the plug does nothing but stabilize the plug, if that's the case, its still safe to operate ungrounded. But a plug that's getting hot should be remedied (replaced). When you do that you'll also have your answer to wether the ground pin is needed. (if the cable has a ground conductor the ground pin is needed for safety)
You answered your own question, the plug is getting really hot NOW. Yes you need to replace the plug. The ground is there for your protection if the vac where to get a direct short and/or a static buildup charge from the carpet.