Does Throwing an Object Overboard Affect Pond Water Level?

[TomK]

Homework Statement

Archimedes' Principle.

Relevant Equations

upthrust = weight of water displaced
density = mass/volume
volume of object submerged = volume of water displaced
mass x g = weight

I was asked this question:

Assume you're sitting in a boat (you and the boat, together, are a mass M) which also contains a heavy object (of mass m). The boat (inc. you and the heavy object) is floating in a pond (a fixed body of water, rather than open sea). If you throw the object overboard (it sinks) and remain seated in the boat, will the water level of the pond rise, fall, or stay the same?

Below, I have given my proposed answer, and I wanted to ask if it made sense:

When the object is in the boat: weight of displaced water = (M+m)g = (M+Vd)g
This is because masses M and m are both floating.

volume x density = mass
d = density of object
V = volume of objectWhen the object is thrown overboard: weight of displaced water = (M+1000V)g
This is because the volume of submerged object = volume of water displaced by object.

Assumes density of water = 1000kgm^-3Comparing the two equations: We know d (density of object) is greater than 1000, because the object sinks in water.
When the object is thrown overboard, the weight (and volume) of displaced water decreases.
Therefore, the water level must have fallen.

Is this the correct conclusion to make? Could you prove this scenario in-practice, or would your boat not stay afloat with the object in it?

 

[haruspex]

Is this the correct conclusion to make?

The reasoning is correct.

would your boat not stay afloat with the object in it?

Do you think that having an object denser than water in a boat would necessarily make it sink?

M+1000V

Better to write M+ρ_wV, or, if you must substitute numbers, quote units: M+(1000V kg/m³).

 

[TomK]

Thank you for the response. This answers my question.

I think my misconception came from thinking that the object (more dense than water) would have to be sufficiently heavy to displace more water when thrown overboard.

However, my working shows that, under the assumed circumstances, m could really be any value and the water level would still change (though it would have to be heavy to make a measurable difference in real-life).