Archimedes and fish

Question: A barrel is sitting on a set of scales, the barrel contains water that brings the weight of the barrel and the water in it to 200 lbs. The scale confirms this weight. I place a fish that weighs 50 lbs in the barrel of water.
1. Does the weight on the scale now increase to 250 lbs.?
2. What if I were to place a 50 lb. block of steel into the barrel, would the combined weight total 250 lbs.?

Answer:

First the simple answers then the explanations and some other points.

1. Yes it will show a weight of 250 lbs.
2. Yes it will show a weight of 250 lbs.

Next a question for you, which I will answer myself. I imagine that the fish is alive and swimming around horizontally.

It actually makes no difference whether the fish is alive or dead (as long as it is floating or swimming around horizontally or indeed whether it is a block of steel.

In each case the object has a weight of 50 lbs (a downward force) and so this has to be supported by something (Newton's Law Three – if a force acts on one body an equal and opposite force acts on another body). In the case of the live fish this upward force is provided by the water but this force is then transmitted through the water to the scales. If the fish is lying on the bottom or if you have your steel weight then the force is provided by the bottom of the barrel. In each case the scales reading will go up by the weight of the object.

(Actually don't dead fish float?)

If you are really interested in this how about thinking about the case where you lower the steel into the water on a piece of string so that you are supporting it.

Because of Archimedes principle there will be an upthrust on the steel which will equal the weight of the water displaced by the steel. The reading on the scales will go up again but only by the amount of the upthrust – you are providing the rest of the support by the tension in the string. (Tension + upthrust will equal 50 lbs using the same block of steel as in your question).

For a floating object the upthrust is equal to the total weight of the object and so the scale reading goes up by the full, weight of the object.

I have used lbs here – strictly they are forces and should be lbs wt or better still the SI unit – Newtons.