Honestly, Coopah, as Steve and Cathy mention, this is not because the plug was plugged in wrong. I suspect the reason the problem went away when you unplugged and plugged it in the other way is that the wires that were shorting together were nudged apart by the movement, and are no longer shorting. This is a
temporary condition,
the cord or the wiring inside MUST be addressed for safety.
The cords were not polarized in the past, because it was not the "standard" or the law. The reason it truly does not matter is that with alternating current - AC power - what's supplied by the power company to your house, there is no positive and negative, hence no right or wrong way to plug the cord in. (Also the reason a motor like this doesn't run in reverse if plugged in one way or the other, or if we switch the wires at the machine side.)
http://en.wikipedia.org/wiki/Electrical_polarity
The other thing to note about AC power, is that it requires a very small amount of it to disrupt the human heart. Less than 1 amp. Your wall plug is likely capable of supplying between 15 and 20 of those amps.
Typically, with 120volt service (what your wall plug is using, but not your dryer), you will be "thrown clear" in an electrocution situation, but it's still not good for your heart.
This is not to be played with. I'm sorry to be sounding "doom and gloom" but it's dangerous to believe that which way you plug it in will make the difference. It would likely be mandatory to change a plug out to a polarized one if simply plugging it in "wrong" could cause electrocution and fires.
Please examine the wiring inside the machine, at the points that were mentioned, and the cord as well, between the foot and the machine as well as from the machine to the wall, and let us know what you find.