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I bought a vintage alarm clock and radio. It works perfectly, but it took me a day to discover that along the length of the cord, there is a small nick, which revealed a small bit of copper wire, which doesn't seem frayed at all.

The cord is unpolarized, and plugs into 120V AC without any transformer "box".

Before the nick was discovered, the clock ran perfectly for a day and a half before unplugging.

I bought a spool of 3M Super 88 electrical tape and thinking of wrapping it up.

Would that be enough, or would it be still unsafe?

That cable, as is, will tend to "bite people". It also has damage that could increase resistance there causing that spot in the cable to get hot. Continued flexing will worsen the wire damage and the heating.

An electrical-tape repair will work temporarily, but I would not resell it like that.

The right way is take your time and find a gray power cord of very similar style, open up the clock and replace the cable.

It depends. The proper procedure would of course be to replace the cord, but from a safety perspective, insulating tape will work. It may be against regulation, and if you have kids or pets around it may be unadvisable.

The main problem is that tape tends to sag and move over time. A better solution than electrical tape would be either zipper tube, which is heat shrink tube that is closed with a zipper, or vulcanizing tape.

Zipper heat shrink tube without glue will provide a lasting insulation. You can even split the conductors, insulate them with a heat shrink tube each to their own, and then an overall shrink tube to keep it together. If you need waterproof (you probably don't), the tubes with thermoglue will provide that.

Same goes for self-vulcanizing tape. It will provide a perfect seal that doesn't sag or slip. You could insulate each conductor individually to ensure that there will be no short circuit.

Either of these are likely to cost as much as, or more, than replacing the cord. Either would be entirely fine from a safety perspective.

Insulating tape will make it safe in the short term. In the long term (or if accessible to pets or children) it can come off or be pulled off.

It is often hard to replace the power cord. Sometimes the case of the appliance is impossible to open. Other times the cable restraint/grommet is moulded onto the appliance end, and no other piece of cable could be secured.

I'd cut the plug off the other end (if moulded on), slide heat-shrink tubing along to cover the nick, and shrink it into place. Then attach a re-wireable plug. Self-amalgamating tape is another possibility, but in my experience it goes gooey with age. Not necessarily unsafe, but not nice. A third possibility is an in-line cable joiner. Cut the cable at the nick, and join the conductors inside the in-line joiner. Some designs of joiner you could simply put the damaged cable inside the body of the joiner and secure it on both sides of the nick without actually cutting the cable at all.

Heat-shrink tube is by far the neatest.

None of this is advising whether these measures are legal under any particular wiring code. Just what I'd regard as making it safe, bearing in mind it's a clock, not, say, a hedge-trimmer that gets carried around and used outside in a wet environment.

Never take chances with electricity. Remove the damaged device from those who might use it. Find the correct gauge cord and replace the entire damaged cord.

What you have there is free access (50% of the time) to the live / hot conductor for a passing finger or other part of your anatomy. If you have an earth leakage device (RCD, might have a different name in the USA which 120V points to being your country) in your mains supply, you'll cause it to trip if you touch it and are grounded... turning off the power to that circuit. It'll hurt. If you don't have an earth leakage device the hurt could be fatal.

Proper insulating tape will work as a makeshift repair but you shouldn't rely on tape for a long-term repair to exposed wiring - the adhesive may perish, the over-wrap may slide up or down the cable, children might unwrap it to see what was underneath. Replacing the cable is a good option but will involve opening the machine and may need soldering. Shortening the cable is a good option. Cutting the cable at the damage and using a cable joiner is a safe but less good option... it depends where the fault is along the length of the cable.

If you're sure the wiring inside the cord isn't damaged, you can probably make a simple repair to the insulation. Zip cord is typically made of a vinyl (poly-vinyl chloride) material and you may be able to find a solvent, like for PVC "plastic" plumbing, that would dissolve the insulation enough to reform it to seal the tear, if there isn't a chunk missing. If there is material missing, perhaps a vinyl repair product would also work.

Since you made a point of mentioning it was vintage, I think these suggestions will give you the most attractive resolution.

I've seen chewed cords and this one looks like it had been caught under a piece of furniture or closed in a drawer. The stretching doesn't suggest teeth marks.

Leaving the cord alone shouldn't affect the operation of your device. Rather, it's a safety issue. Electrical tape would prevent contact with the wiring, but it's not an attractive option. Using a vinyl adhesive or patching product would restore the integrity of the insulation without compromising aesthetics.

If the wiring is actually damaged, though, replacement or shortening the cord to removed the damaged area is best.