Hydrostatic Pressure on Curved Circular Surface

[Tom Hardy]

Homework Statement

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The Attempt at a Solution

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Can someone please help..There's no answers and I think my answer is very wrong, as you can see I have pretty much no idea what I'm doing.

Thanks

Edit:

Where it says net force I meant to say net force per unit width.

Edit2: My integration is wrong, it should read Density*gravitational constant*R^2(-1+cos(sin^-1(h/R)))

 

[Chestermiller]

Hi Tom,

The pressure at a given depth is isotropic, meaning that it acts equally in all directions (i.e., perpendicular to any surface of arbitrary orientation). You correctly determined the force per unit width between Θ and Θ+dΘ as $\rho g z Rd\theta$, where z is the local depth. This pressure force is acting perpendicular to the surface of the gate at depth z. So, you need to resolve this local force into components in the horizontal and vertical directions. You then integrate the horizontal and vertical components over the wetted surface of the gate.

Chet

 

[Tom Hardy]

Hi Tom,

The pressure at a given depth is isotropic, meaning that it acts equally in all directions (i.e., perpendicular to any surface of arbitrary orientation). You correctly determined the force per unit width between Θ and Θ+dΘ as $\rho g z Rd\theta$, where z is the local depth. This pressure force is acting perpendicular to the surface of the gate at depth z. So, you need to resolve this local force into components in the horizontal and vertical directions. You then integrate the horizontal and vertical components over the wetted surface of the gate.

Chet

Hi Chet,

Thank you for replying, I appreciate it a lot. Does this solution look better to you? (the first F/width is in the x direction and the one after in the y direction).

 

[Chestermiller]

Hi Tom,

This solution really isn't correct. But I think I can help you to arrive at the correct solution by asking you a sequence of leading questions.

At a given angle Θ, what is the depth of the location on the gate below the surface (in terms of R, h, and Θ)?

After you answer this, more questions to follow.

Chet

 

[Tom Hardy]

Hi Tom,

This solution really isn't correct. But I think I can help you to arrive at the correct solution by asking you a sequence of leading questions.

At a given angle Θ, what is the depth of the location on the gate below the surface (in terms of R, h, and Θ)?

After you answer this, more questions to follow.

Chet

Thanks for sticking with this, I thought h was the local depth (variable?), maybe I misunderstood the diagram. If we call h is a constant (representing the distance between the crest level and the surface of the water) then the local depth would just be z = h - Rsin \Theta
So the force per unit width acting perpendicular to the gate at d \Theta would be \rho gzRd \Theta = \rho gR(h-R \sin \Theta) d \Theta is that correct? If so, should I resolve that into it's horizontal and vertical components and then integrate from there?

 

[Chestermiller]

Tom! You're the man! Excellent!

The answer to your question is yes.

Chet

 

[Tom Hardy]

Tom! You're the man! Excellent!

The answer to your question is yes.

Chet

Awesome, thanks so much for your help, I really appreciate it.

 

[Tanya Sharma]

Hello Chet

Should the horizontal force per unit width on the curved surface be $\frac{1}{2}ρgh^2$ ?

 

[Chestermiller]

Hello Chet

Should the horizontal force per unit width on the curved surface be $\frac{1}{2}ρgh^2$ ?

Hi Tanya.

Yes. That's what Tom's integral comes out to be. And, of course, the answer could have been obtained much more easily in another way (which is the way, I'm sure, that you thought of). But I thought it would be useful for Tom to continue evaluating it by the integration. Would you like to reveal for Tom the much simpler way you figured it?

Chet

 

[Tanya Sharma]

Please have a look at the attached figure . The cylinder is lying on a frictionless surface with liquid of density 2ρ on left and liquid of density 3ρ on right . The cylinder is in equilibrium and our task is to find 'h' .

I am having problem applying this result to the situation in the figure.I know how to solve this problem wihout much maths,but would like to solve it with the above mentioned integration approach .

Let h = x +R

Using the result in post 8 , doing the horizontal force balance on the cylinder

$\frac{1}{2}(2ρ)gx^2 + \frac{1}{2}(2ρ)gR^2 = \frac{1}{2}(3ρ)gR^2$

This gives , $x = \frac{R}{\sqrt{2}}$ and $h = R + \frac{R}{\sqrt{2}}$ .

But value of h should come out to be $\sqrt{\frac{3}{2}}R$ .

Could you help me in figuring out the mistake ?

 

[Chestermiller]

Please have a look at the attached figure . The cylinder is lying on a frictionless surface with liquid of density 2ρ on left and liquid of density 3ρ on right . The cylinder is in equilibrium and our task is to find 'h' .

I am having problem applying this result to the situation in the figure.I know how to solve this problem wihout much maths,but would like to solve it with the above mentioned integration approach .

Let h = x +R

Using the result in post 8 , doing the horizontal force balance on the cylinder

$\frac{1}{2}(2ρ)gx^2 + \frac{1}{2}(2ρ)gR^2 = \frac{1}{2}(3ρ)gR^2$

This gives , $x = \frac{R}{\sqrt{2}}$ and $h = R + \frac{R}{\sqrt{2}}$ .

But value of h should come out to be $\sqrt{\frac{3}{2}}R$ .

Could you help me in figuring out the mistake ?

Your error is on the left hand side of the equation. It should read: $\frac{1}{2}(2ρ)g(R+x)^2=\frac{1}{2}(2ρ)gh^2$

Chet

 

[Tanya Sharma]

I understand what you are saying , but I think i haven't explained myself properly .

I will go a bit slowly in order to find the mistake . Isn't the horizontal force exerted by fluid on the left side from bottom till height R = $\frac{1}{2}(2ρ)gR^2$ ?

 

[Chestermiller]

I understand what you are saying , but I think i haven't explained myself properly .

I will go a bit slowly in order to find the mistake . Isn't the horizontal force exerted by fluid on the left side from bottom till height R = $\frac{1}{2}(2ρ)gR^2$ ?

No it isn't. What is the pressure at height R?

Chet

 

[Tanya Sharma]

Thanks .

I have found the error in the maths involved in the problem . Now I would like to understand it a bit conceptually .

If the liquid on the left side were up to height R only , then the horizontal force would have been $\frac{1}{2}(2ρ)gR^2$ . Right ?

But this changes (i.e force due to liquid from bottom to height R ) from $\frac{1}{2}(2ρ)gR^2$ to $\frac{1}{2}(2ρ)g(R^2+2xR)$ , if the liquid on the left has height h = x +R , x ≥ 0 .

Could you explain this qualitatively.

 

[Chestermiller]

Thanks .

I have found the error in the maths involved in the problem . Now I would like to understand it a bit conceptually .

If the liquid on the left side were up to height R only , then the horizontal force would have been $\frac{1}{2}(2ρ)gR^2$ . Right ?

But this changes (i.e force due to liquid from bottom to height R ) from $\frac{1}{2}(2ρ)gR^2$ to $\frac{1}{2}(2ρ)g(R^2+2xR)$ , if the liquid on the left has height h = x +R , x ≥ 0 .

Could you explain this qualitatively.

So the real question is, "what does $\frac{1}{2}(2ρ)g(2xR)$ represent physically?"

What if we wrote this in a slightly different way: $[(2ρ)gx]R$? What does $[(2ρ)gx]$ represent physically?

Chet

 

[Tanya Sharma]

No . This isn't what I intended to ask . Sorry for not being clear with my questions .

Nevertheless , I have found the error in my reasoning .

Thank you very much . I am glad I discussed my doubts with you .