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If you have a cork piece on top of a nail, it is extremely hard to keep it stable, and the slightest action will make the cork fall off. However, when you balance it on top of a nail but put forks into it, apparently it becomes extremely easy to keep stable. I looked in my textbook and it said that this is because the center of mass is lowered from its original position (and the center of mass was below the point of contact in the first place), which is confusing because if you are adding more mass towards the top, shouldn't the center of mass move upward and not down? It also says that when you try to topple the cork with the forks in it, the center of mass moves upward, increasing its potential energy.

Can someone really good at this stuff explain the entire process to me? I am truly confused about this.

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The experiment involves sticking the tines of the forks into the cork so that the long heavy handles of the forks extend downward. Take a look at the photo in this link: https://www.kecksci.claremont.edu/physics/demo/corkfork.htm.

Now, the cork and the forks are bound together as one object, and the center of mass of that object is down toward the middle of the forks, well below the top of the cork. You've effectively lowered the center of mass of the cork by making it part of a cork/forks object. This link explains the process: http://scienceblogs.com/principles/2007/12/17/the-twofork-toothpick-trick-ex/, though using a toothpick rather than a cork.

With the center of mass lower than the point of contact, gravity works to stabilize the object, because if the object tilts (raising the center of mass), the center of mass wants to drop down again and the object will automatically balance.

But if the center of mass is higher than the point of contact, gravity tends to de-stabilize the object because any slight movement lowers the center of mass, beginning the process of falling.

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The reason for stability can be explained in terms of potential energy.
Systems want to reach a state of minimum potential energy.
Assume a system that is in static equilibrium undergoes a small displacement from that position.
If such a displacement results in the potential energy increases the system will try and go towards the stable equilibrium position to lower its potential energy. The opposite is true if the small displacement results in a decrease in potential energy.

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