Statics problem about an anchored plank in water

[vanillin]

Hello, I'm new and quite unfamiliar with all the conventions, but I'll use underline for vectors since I couldn't find the proper thing (is the convention to use bold here?)

Problem:
No more.Relevant equations:
F_net = ma = 0
τ_net = Iα = 0
τ = r x F = rFsinθ
F_buoyancy = ρ_waterVgAttempt at solving:
What I realized is that since the plank isn't in translational nor rotational motion, the net force and torque are both zero. T = tension, G = gravitational f., B = buoyancy.

T + G + B = 0

in scalar:

T + G - B = 0

Also, I know how to equate the torques (only gravity and buoyancy cause torque), but I can't find out the distances nor angles. I can also express the gravitational force in terms of density of the plank and it's volume...? Things just don't cancel out neatly there.

I can provide a diagram if really necessary, but I'd prefer not to now, I have very limited time. Thanks in advance. Also, is the template there to be followed rigorously or is any kind of organised posting viable?

 

[LawrenceC]

Hint:

Compute center of pressure as a function of angle and enforce equilibrium using moment summation.

 

[vanillin]

Hint:

Compute center of pressure as a function of angle and enforce equilibrium using moment summation.

I would really like to have a solution given, or at least all the steps necessary, because I've been fighting with this for a good while and I don't feel like anymore, as it's not the only thing I need to know for Wednesday.

If nobody gives me the answer before tomorrow, I might try it then... but I would really appreciate a solution, as I've tried to do this.

E: Just noticed that complete solutions violate the rules... well is it possible to at least give me the steps, and I could do them myself?

 

[LawrenceC]

Find the upward force due to the portion under water as a function of the angle and the distance from the surface. That force is directed vertically and is applied at the center of gravity of the portion under water. This creates a moment. Determine the moment in the other direction due to its natural weight. The plank is in equilibrium so what do you know about the sum of the moments?

 

[vanillin]

Find the upward force due to the portion under water as a function of the angle and the distance from the surface. That force is directed vertically and is applied at the center of gravity of the portion under water. This creates a moment. Determine the moment in the other direction due to its natural weight. The plank is in equilibrium so what do you know about the sum of the moments.

L = length of plank

the angle θ between the forces and the moment arm is the same for both as the forces are only antiparallel.

B = ρ_waterV_submergedg
G = ρ_plankV_plankg

V = ?

τ_B = rBsinθ = (L/2 - x) * ρ_waterV_submergedg sinθ
τ_G = rGsinθ = (L/2) * ρ_plankV_plankg sinθ

(L/2 - x) * ρ_wV_subg sinθ = (L/2) * ρ_pV_pg sinθ

→ (L/2 - x) * ρ_wV_sub = (L/2) * ρ_pV_p

Sum of the moments is zero, that I knew. I can identify all the forces and where they act, I just cannot make them into reasonable equations. Like volume, how do I get rid of it?

 

[LawrenceC]

The volume consists to a cross section multiplied by length. The cross section is arbitrary but appears in each term so it cancels.

 

[vanillin]

Oh, foolish me... I was stuck at the T + G - B = 0 with no hope of getting rid of V or A... had I only started with the moments.

Thanks, I think it makes sense now, I'll try it and see if I get the answer.

E: Yes, I got the answer, thank you. I think the angle is a matter of geometry.