Engineering Hydrostatic Pressure of Ball Floating in Liquid

[hayleejo34]

Homework Statement

An object floats in liquid. The depth of the object below the surface is d and its height above the water is h. The density of the ball is p_b and the density of the liquid is p_liq. What is the pressure at the bottom of the object? The pressure above the water is P_atm.

Relevant Equations

hydrostatic equation: dP/dz = -pg

Let's say there's an object floating in liquid. The depth of the object below the surface is 𝑑 and its height above the water is ℎ. The density of the ball is 𝑝𝑏 and the density of the liquid is 𝑝𝐿. The pressure above the water is 𝑃𝑎𝑡𝑚.

I understand this situation when only the density of the ball matters; you integrate to get P(z) = 𝑝𝑏𝑔𝑧 + C, and given the initial condition P(d) = 𝑃𝑎𝑡𝑚. (pressure at water-air interface is atmospheric), C = 𝑃𝑎𝑡𝑚 - 𝑝𝑏𝑔𝑑 , so P(z) = 𝑝𝑏𝑔𝑧 + 𝑃𝑎𝑡𝑚 - 𝑝𝑏𝑔𝑑

What I am confused about is how to find the pressure given the influence of the density of the liquid. I don't really understand why or how the density of the liquid affects the equation for pressure at all, besides that in order for the object to be floating, 𝑝𝑏< 𝑝𝐿. It seems like since you're only interested in the ball, so the only distance that matters is the distance of the ball under the water, and the only density relevant there would be the one of the ball. Or do you take the ratio of the densities? Is it their difference? Why does the liquid matter at all?

 

[Lnewqban]

Welcome!
I believe that the pressure at the bottom of the floating ball is the same for all the points located along a horizontal line at a d distance under the surface of the liquid.

 

[hayleejo34]

Welcome!
I believe that the pressure at the bottom of the floating ball is the same for all the points located along a horizontal line at a d distance under the surface of the liquid.

Hey, thanks! That part makes sense! But how does the density of the liquid relate to the density of the liquid in the hydrostatic equation?

 

[Lnewqban]

Hey, thanks! That part makes sense! But how does the density of the liquid relate to the density of the liquid in the hydrostatic equation?

Please, see:
https://www.physicsforums.com/threads/weight-of-fluids-on-a-submerged-person.992756/