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A drop in the ocean

Posted by ubpdqn on February 28, 2012

Professor Lewin gave the  following puzzle on his facebook site:

You are sitting in a boat which is floating in a small swimming
pool. No water is flowing in or out of the swimming pool. There is
also a big (heavy) granite rock in your boat. The surface of the
water is a perfect horizontal flat plane; there is no wind, there are no ripples on the water.

While you and the rock are floating in the boat, you place a mark (A)
somewhere on the wall of the swimming pool PRECISELY at the surface of the water.

You now throw the rock over board. The rock sinks and ends up on the
bottom of the pool. You wait a little till the waves that you caused are all gone. Thus the water is again a completely flat horizontal plane. You now look at your mark A.

Question: Will the surface of the water now be
1. above A
2. below A or
3. again exactly at A?

This is an exercise in the understanding of Archimedes principle. For floating objects (average density less than water), the volume of water displaced is equal to the mass of the floating object (clearly the volume displaced is less than the volume of the floating object or else the object would sink). When the object sinks, the displaced volume is the volume of the sinking object.

As a a thought experiment, initially the displaced volume is the combined mass of rock and non-rock. After the rock has sunk the displaced volume is the volume related to the mass of the floating non-rock (Archimedes principle) plus the volume of the rock. As the rock is denser than water the volume of the rock is less than the volume of water of the same mass as the rock (i.e. the volume that was displaced while it was floating in the boat). Hence, after the rock is displaced the volume of water displacement is less than the initial state.  The water level drops.

In a somewhat messy manner, I confirm and quantitate this in the following:

I preliminary CDF is posted here. It is rather contrived. The effect on boat top position is dependent on the particulars I chose for the demonstration.

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