Telephone Pole Partially
Submerged in a Lake

Problem Diagram

A wooden telephone pole paritally falls into a lake as shown. What is the density of the pole? Assume the density of the water is 1.94 slug/ft³.

The pole will come to a static position where the sum of the moments about the point O is equal to zero.

     ΣM_o = 0

Therefore, the buoyancy force multiplied by its lever arm must be equal and opposite the gravitational force times its lever arm.

     ΣM_o = 0 = (W_poleL_pole - F_BL_B)cosθ

where

L_pole = L/2 (gravity acts uniformly along the entire length), and

(buoyancy force acts at the centroid of the submerged part of the pole). Note, θ is not needed since it factors out of the equation. Summing moments at O gives,

     W_poleL_pole = F_BL_B

where

     W_pole = ρ_pole V g = ρ_pole (π r² L) g

     F_B = ρ_water V g = ρ_water (π r² L/3) g

thus,

      ρ_pole (π r² L) g (L/2) = ρ_water (π r² L/3) g (5L/6)

Cancelling terms,

     ρ_pole (1/2) =(1.94 slug/ft³) (5/18)

The density of the pole is

     ρ_pole = 1.078 slug/ft³

Septic Tank

Septic tanks are used by homes in rural areas not serviced by sanitary sewers. Assume the tank is 6 ft x 6 ft x 10 ft (long) with 4 in. concrete walls, and assume 1 ft of soil over the tank. At what groundwater depth, d, if any, will the tank float? Any recommendations to keep it from floating? Assume γ_concrete = 150 pcf, γ_soil = 120 pcf, and γ_water = 62.4 pcf

Inside dimensions: h, w, l
Outside dimensions: h + 2t, w + 2t, l + 2t
where t is the wall thickness

     V_c = (h + 2t)(w + 2t)(l + 2t) - hwl
     A = (w + 2t)(l + 2t)

Since there is 1 ft of soil over the top of tank,

     z = (1 ft) + (h +2t) - d

where d is the water table.

     γ_concrete = 150 pcf
     γ_soil = 120 pcf
     γ_water = 62.4 pcf

To float, the buoyancy force acting on the bottom of the tank must be greater than the weight of the tank and the soil on top.

septic tank: h = 6 ft, w = 6 ft, l = 10 ft, t = 4 in.

     V_c = 474.1 - 360 = 114.1 ft³
     A = 71.11 ft²
     z = 7.67 - d

If the groundwater table depth is less than 1.89 ft, the septic tank will float.

Recommendation to keep it from floating: use 6 in. walls instead of 4 in.