Fear that someone else would build them first and attack the US or our allies.
During World War 2 the "someone" was Nazi Germany and the nuclear weapon was the atomic fission bomb, during the Cold War the "someone" was the USSR and the nuclear weapon was the hydrogen fusion bomb.
The US clearly won the race with Nazi Germany as their project derailed early due to multiple errors and oversights by their scientists and the work was redirected away from bombs to prototype reactors, however the US and USSR both developed and fielded practical megaton range hydrogen bombs in the same year: 1955, making that race effectively a tie!!!
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Because there was no true defense against nuclear weapons. Once both sides had nuclear weapons, the only way to "protect" themselves from the other side was to have so many more nuclear weapons that even if their enemies used all of their nuclear weapons, there would still be nuclear weapons to shoot back with. That way, nobody would use nuclear weapons, because they could never actually "win" that way. This thought process was referred to as Mutually Assured Destruction, or MAD.
Fear that the North Vietnamese would obtain nuclear weapons themselves (from Russia or China) and use them on South Vietnam or even on America.
to attck the countries that hurt the u.s , such as hiroshima. even though they are very destructive.
Nuclear weapons serve as a deterrent. One country will be less likely to attack another that it knows has nuclear capability because of the destruction that could be wrought by nuclear weapons. This theory is called MAD or Mutually Assured Destruction, meaning that both sides would destroy each other so utterly by going to war that peace (even bitter cold peace) is the only viable option. Biological and Chemical weapons, which are also called weapons of mass destruction, do not have this deterrent effect.
Usually the quantity of fissile material must be accounted for to the milligram, even though there are tons of it in stock.