How Does the Normal Force on a Rock in Water Differ with Varying Water Levels?

[phrygian]

Homework Statement

A rock of mass M with a density twice that of water is sitting on the bottom of an aquarium tank filled with water. The normal force exerted on the rock by the bottom of the tank is?

Homework Equations

The Attempt at a Solution

The answer is Mg/2, I understand how to arrive at that by subtracting the buoyant force which is equal to mg from the gravitational force on the object which is 2mg. but shouldn't the normal force on the object be different when there are different amounts of water pushing down on it? Since the amount of water is not given shouldn't the correct answer be "impossible to determine from the information given?

 

[Q_Goest]

Hi phrygian. Are you familiar with the Archeimedes' principal? Let's not talk about "amount" of something. Let's give everything a specific, measurable quantity such as "volume" or "mass". Archimedes' principal simply states that any object, wholly or partly immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object.

So the rock displaces some volume of water which has a given weight.

Does that help?

 

[phrygian]

I understand that but if I make a free body diagram for the rock wouldn't the forces be normal force + buoyant force - gravitational force - force due to pressure from the fluid above = 0 ? So doesn't the normal force change as the force due to pressure from the fluid above changes when there are different amounts of fluid above?

 

[Q_Goest]

A free body diagram of the forces on the object in the water would consist of the forces down (weight of object) minus the forces up (integrate pressure over the area). This can be simplifed if we imagine a cubical volume. Forces on the sides cancel out. Force on top equals pressure times area. Force on bottom equals pressure times area. Use Bernoulli's equation to determine pressure. The force up (difference in pressure times area) minus the force down (weight) equals the weight of the fluid, which is exactly what Archimedes' principal tells us.