Calculating Gauge Pressure on a Sphere

LoveBoy · Mar 25, 2016

Homework Statement

Diagram of a metal ball of radius r lying on the bottom of a container filled with liquid to a depth of 4r, with the liquid surface and a height marker of 4r labelled, plus four multiple-choice answers involving pi, r squared, P0 and rho g.

Homework Equations

What is gauge pressure ?

The Attempt at a Solution

Two typeset equations: first, Force = Pressure × Area; second, Force = P₀(πr²) + gauge pressure.

How to find gauge pressure ?

LoveBoy · Mar 25, 2016

@Nathanael can you help ?

Nathanael · Mar 25, 2016

Gauge pressure is the true pressure minus the ambient air pressure. It can be thought of as "the pressure difference compared to the surface of the liquid." (It is zero at the surface of a liquid. ) Maybe another way to say it is that it is the pressure due only to the weight of the liquid (as opposed to the liquid+atmosphere).

It doesn't make sense to me why they said "Find the force (...) due to gauge pressure" because all of the answers involve the atmospheric pressure... so they are obviously not using the gauge pressure.

I would say just pretend it says "pressure" in place of "gauge pressure."

LoveBoy · Mar 25, 2016

Nathanael said:

Gauge pressure is the true pressure minus the ambient air pressure. It can be thought of as "the pressure difference compared to the surface of the liquid." (It is zero at the surface of a liquid. ) Maybe another way to say it is that it is the pressure due only to the weight of the liquid (as opposed to the liquid+atmosphere).

It doesn't make sense to me why they said "Find the force (...) due to gauge pressure" because all of the answers involve the atmospheric pressure... so they are obviously not using the gauge pressure.

I would say just pretend it says "pressure" in place of "gauge pressure."

Thanks for your explanation .
Can you please give me some more hint ?

Nathanael · Mar 25, 2016

LoveBoy said:

Can you please give me some more hint ?

I don't want to hint at it too much before you attempt the problem, but I will say this much:

There are two approaches (that I see), the messy approach and the neat approach. The messy approach involves integrating the pressure over the surface area. The neat approach involves using archimedes principle in a clever way. I will let you contemplate on how to use archimedes principle to solve this problem. (The less hints I give the more satisfying it will be to solve!)

LoveBoy · Mar 26, 2016

Well , I'm sorry , i can't calculate the answer yet.

ehild · Mar 26, 2016

LoveBoy said:

Well , I'm sorry , i can't calculate the answer yet.

Cut the sphere into halves, and remove the bottom half. Consider the forces acting on the upper half sphere . The force Fb acting at he base is upward, and there is the downward force Fs on the upper half spherical surface. The result of these forces is the buoyant force, B. You can determine Fb and B, so you get Fs.

Free-body diagram of a hemisphere of radius r submerged in a tank of liquid, with the lower half removed and shown dashed, downward force Fs on the curved top surface and upward force Fb on the flat base, and the tank depth labelled 4r.

LoveBoy · Mar 27, 2016

Handwritten-style typed equations: Fb = ρVg(↑), B = dVg(↑) where d = density of upper hemispherical part, So, Fs = (Fb + B)(↓).

Am i right till here ?

ehild · Mar 27, 2016

LoveBoy said:

Am i right till here ?

Not yet. What is V?
Fb is the force exerted by the fluid on the base of the semiphere. It is equal to the hydrostatic pressure at the position of the base multiplied by the area of the base. At what depth is the base of the semiphere? What is the hydrostatic pressure there?
B is the buoyant force - it is equal to the weight of the fluid displaced.How much is it?

LoveBoy · Mar 27, 2016

Handwritten blue annotation beside typed text: F_b = ρVg is set equal to handwritten P×A; below, 'where V = Volume of other hemisphere' and 'Hydrostatic pressure at the base of hemisphere = 3ρgr'.

ehild said:

B is the buoyant force - it is equal to the weight of the fluid displaced.How much is it?

By the way,i'm confused in finding the buoyant force i.e weight of fluid displaced .

ehild · Mar 27, 2016

LoveBoy said:

By the way,i'm confused in finding the buoyant force i.e weight of fluid displaced .

In my notation, Fb is the force exerted by the fluid at the base of the upper hemisphere. It is equal to the pressure at that depth multiplied by the area of the hemisphere. The radius of the sphere is r, what is that area then? And what is Fb?

Do you know Archimedes' principle? The buoyant force B is equal to the weight of the fluid displaced by the upper hemisphere. There is no fluid at the place of the hemisphere, so the volume of the displaced fluid is equal with the volume of the hemisphere. If you know the volume and the density, you can get the weight, don't you?

LoveBoy · Mar 27, 2016

Typed equations: buoyant force Fb = (ρgr)(A) = (ρgr)(πr²); displaced fluid equals volume of the hemisphere; weight of displaced fluid = density × volume = (d)(2/3 πr³), where d = density of sphere.

Am i right now ?

ehild · Mar 28, 2016

LoveBoy said:

Am i right now ?

The pressure at a depth is proportional to the depth. The base is at depth of 3r. What is the hydrostatic pressure then?
The pressure in the fluid depends also on the atmospheric pressure Po. The pressure adds to the pressure of the fluid, so the pressure at the base is Po+3rρg. The force exerted on a plane surface of area A is independent of direction of the surface. It is the same upward or downward or sideways.

Diagram of a solid hemisphere of radius r submerged flat side down in a fluid-filled container, with arrows labeled Po at the top surface, Fs pointing down on the curved surface, and Fb pointing up at the base; depth labels 3r and 4r.

It is correct that the volume of the displaced fluid is (2/3)πr³, but the weight of the fluid is the volume multiplied by the density of the fluid and g .
The buoyant force comes from the difference of the upward force, acting at the base of the hemisphere and the downward force exerted on the spherical surface.

LoveBoy · Mar 28, 2016

Handwritten-style equations: Fs = (P0 + 3rρg) × (surface area), where surface area = 3Πr^2; Fb = (2/3) Πr^3 ρg; B = Fs − Fb.

Is it correct ?

ehild · Mar 28, 2016

Your notations are completely different from my ones. Is B the force on the spherical surface? And is Fb the buoyant force? And is Fs the force on the base of the hemisphere? If yes, you are right. What do you get for B?

LoveBoy · Mar 28, 2016

Thanks a lot @ehild !
I got the answer !

Calculating Gauge Pressure on a Sphere

Homework Statement

Homework Equations

The Attempt at a Solution

Discussion

Similar threads