Cleaning pennies with salt and vinegar is a classic elementary school science experiment. Using the same principles, and a little patience, it is possible to dissolve a penny completely. When cleaning a penny, the hydrochloric acid produced by the mixture of salt and vinegar dissolves a thin layer of copper on the penny. Repeatedly allowing copper oxide (the green stuff that looks like dirt on the penny) to form and "cleaning" it will slowly but surely reveal a quick-to-dissolve zinc core.
Choose a penny dated after the year 1982. This is the year that the government began making pennies with a zinc center. Zinc is a more reactive metal than copper, and will produce faster results.
Dissolve as much salt as possible in 8 oz. of white vinegar in a glass container. This produces hydrochloric acid. The more salt you can dissolve, the more acid you will produce, and the better your result.
Drop the penny in the acid, handling it with a pair of tweezers. Allow the penny to react until it is clean, then use the tweezers to remove it from the acid and place it on a paper towel. Do not rinse the penny.
Wait until the penny reacts with the air to form copper oxide (the green substance coating the penny). When the penny is coated, use the tweezers to place the penny back in the acid. Wait until the copper oxide fully dissolves, then remove the penny again to allow more copper oxide to form.
Continue to submerge and remove the penny until the copper dissolves, revealing the zinc interior. Once the zinc is visible, leave the penny in the acid. The zinc will continue to react with the acid until the penny is dissolved.