newtonian mechanics - Will a bathtub full of water weigh more if I add something that floats in the water?

Let's say I have a bucket or something bigger, like a bathtub full of water. I weigh that bath and get some value, $x$.

Then I add a small boat made of wood to that bathtub that doesn't touch the borders (i.e., it is only touching the water).

Will this boat add to the total weight of the bathtub, or does the weight of the tub stay the same?

Answer

Mass is mass. If you add something which has mass to the bucket, the bucket now has more mass. It doesn't matter if it was more dense or less dense. If you add 1kg to a bucket, you add 1kg.

Now there are two areas where this falls apart. One is in the case where the object you are adding is actually less dense than the air (not just less dense than the water, but actually less dense than air, like a helium balloon). In this case, we can forget about the minor detail of whether the object is floating on the water or not, we can focus on the entire bucket which is immersed in air. This object displaces a larger mass of air than its own mass, so it will actually add lift to the bucket. The mass of the bucket will still be bucket+water+object, but the bouyancy forces will make the bucket feel lighter. Indeed, this is precisely the mechanism used by hot air balloons.

The other corner case is the case where the bucket is already filled to the brim with water. Adding a single drop of water would cause a drop to have to spill over the brim. Now when we add our toy boat the situation gets a little more complicated. The boat will displace a mass of water equal to the mass of the boat. This displacement will cause the water to rise and spillover. If you tally up all of the masses in this case, you will find that the bucket+water-in-the-bucket+toy boat will have the same mass as the bucket+water did before you added the toy boat. Where did the extra mass go? There's a puddle on the ground outside of the bucket, whose mass is exactly equal to that of the toy boat.