How Do Water Height and Gate Angle Relate in Hydrostatic Calculations?

[RAP1234]

Homework Statement

A 3 m long gate of weight 4 kN per unit width is hinged at O and sits at an angle θ as a function of water height h above the hinge. (A) Using a y-axis measured up from the hinge, derive a general relation between h and θ, with all other variables evaluated in the relation. (ANS: h ∼ C(cosθ sin2 θ) ^1/3 , where you must find C.) (B) Using your result, what h will cause the gate to be inclined at θ = 45◦ ? (ANS: h(45◦ ) ∼ 1.2m)https://www.physicsforums.com/attachments/asd-png.92500/

Homework Equations

The Attempt at a Solution

My answer is not of the form h ∼ C(cosθ sin2 θ) ^1/3I am obligated to use the coordinate system with y increasing from the hinge, up along the gate.
I chose z positive direction going straight vertically upwards from the hinge

dF= PdA eq(1)

dM=ydF eq(2)

dP/dz = rho*g eq(3)Integrating eq(3)

P = rho*g*z+C

but at z = h , P = 0 gage
--> C =rho*g*h

--> P = rho*g*z- rho*g*h\

replacing this into eq(1)

dF = (rho*g*z- rho*g*h) *w*dy

needing z as a function of y, I find that z = ysin(theta)

--> dF = (rho*g*ysin(theta)- rho*g*h) *w*dy

replacing this into eq(2) I find that

dM = (rho*g*y^2*sin(theta)- rho*g*h*y) *w*dy

Integrating the right side with bounds from y=0 to y= 3 I find that

M = w*(9*rho*g*sin(theta)-4.5*rho*g*h) eq(4)

which should be the moment due to the fluid

I thought that since the moment due to the fluid should equal the moment due to the weight of the gate acting n the centroid I would have something like

Mfluid =M weight of gate

Since I am given M due to weight of gate is 4kN per unit width, dividing eq(4) by w

(9*rho*g*sin(theta)-4.5*rho*g*h) = 4,000*cos(theta)*(L/2)

but given in the problem statement is L = 1m

so I replaced in the appropriate values for the density of water rho = 10^3, g = 9.8, L = 1, and solved for h in terms of theta.

But my answer is not of the proper form that problem suggested h ∼ C(cosθ sin2 θ) ^1/3

Please help. I am truly not knowing what I am doing wrong here and don't have anyone I can ask to verify my method.
apologize. I didn't realize the image would be so small when uploaded.
I am obligated to use the coordinate system with y increasing from the hinge, up along the gate.
I chose z positive direction going straight vertically upwards from the hinge

dF= PdA eq(1)

dM=ydF eq(2)

dP/dz = rho*g eq(3)Integrating eq(3)

P = rho*g*z+C

but at z = h , P = 0 gage
--> C =rho*g*h

--> P = rho*g*z- rho*g*h\

replacing this into eq(1)

dF = (rho*g*z- rho*g*h) *w*dy

needing z as a function of y, I find that z = ysin(theta)

--> dF = (rho*g*ysin(theta)- rho*g*h) *w*dy

replacing this into eq(2) I find that

dM = (rho*g*y^2*sin(theta)- rho*g*h*y) *w*dy

Integrating the right side with bounds from y=0 to y= 3 I find that

M = w*(9*rho*g*sin(theta)-4.5*rho*g*h) eq(4)

which should be the moment due to the fluid

I thought that since the moment due to the fluid should equal the moment due to the weight of the gate acting n the centroid I would have something like

Mfluid =M weight of gate

Since I am given M due to weight of gate is 4kN per unit width, dividing eq(4) by w

(9*rho*g*sin(theta)-4.5*rho*g*h) = 4,000*cos(theta)*(L/2)

but given in the problem statement is L = 1m

so I replaced in the appropriate values for the density of water rho = 10^3, g = 9.8, L = 1, and solved for h in terms of theta.

But my answer is not of the proper form that problem suggested h ∼ C(cosθ sin2 θ) ^1/3

Please help. I am truly not knowing what I am doing wrong here and don't have anyone I can ask to verify my method.
[/B]

 

[haruspex]

dP/dz = rho*g

Yes, if you will be giving g a negative value.

Integrating the right side with bounds from y=0 to y= 3 I find that

That extends beyond the region in contact with the water.

9*rho*g*sin(theta)

Looks like you forgot a step in integrating y².

 

[RAP1234]

Yes, if you will be giving g a negative value.
That extends beyond the region in contact with the water.

Looks like you forgot a step in integrating y².

I integrated and evaluated the bounds of the definite integral from y = 0 to y = 3

BUT as you said before the water does not go to the height of the gate.I will fix this and get back to you.

 

[RAP1234]

I am getting an answer of the proper form, but the constant C in front of it I am not sure of. Will the moment due to the weight of the gate have a change in the length of the lever arm since it is partially submerged?

 

[SteamKing]

I am getting an answer of the proper form, but the constant C in front of it I am not sure of. Will the moment due to the weight of the gate have a change in the length of the lever arm since it is partially submerged?

Yes. The net moment about the hinge should be zero for the gate to remain in equilibrium.

 

[RAP1234]

so we would still take 3/2 to be the length of the lever arm, as gravity still acts at the centroid

 

[SteamKing]

so we would still take 3/2 to be the length of the lever arm, as gravity still acts at the centroid

The weight of the gate is fixed, and so is its lever arm about the hinge. The same cannot be said of the hydrostatic force acting on the gate, nor its lever arm reference to the hinge. Those are functions of the water depth h.

 

[RAP1234]

The weight of the gate is fixed, and so is its lever arm about the hinge. The same cannot be said of the hydrostatic force acting on the gate, nor its moment arm.

Thank you. I did fix the integration with the moment portion for the hydrostatic force from 0 t y = h/sin(theta)

 

[haruspex]

so we would still take 3/2 to be the length of the lever arm, as gravity still acts at the centroid

It depends what you mean. The length 3/2m is not orthogonal to gravity.
Please post your new working.

 

[RAP1234]

I posted pictures of my new working below. I do appreciate the help

 

[haruspex]

I posted pictures of my new working below. I do appreciate the help

That all looks pretty good, except that you went wrong early on.
I gather you are taking g as having a positive value, so switched the sign on dP/dz to minus. But that should change the sign on the constant of integration too. Look at your expression for P(z) and consider its value when z=h.

 

[RAP1234]

That all looks pretty good, except that you went wrong early on.
I gather you are taking g as having a positive value, so switched the sign on dP/dz to minus. But that should change the sign on the constant of integration too. Look at your expression for P(z) and consider its value when z=h.

Thanks for the response. I will have to fix that as P should be zero when z = h

Update: Wonderful . After fixing it properly, everything seems to be within range. Thanks again for your help :)