When the Tucurui Dam was constructed in northern Brazil, the lake that was created covered a large forest of valuable hardwood trees. It was found that even after 15 years underwater the trees were perfectly preserved and underwater logging was started. During the logging process a tree is selected, trimmed, and anchored with ropes to prevent it from shooting to the surface like a missile when cut. Assume that a typical large tree can be approximated as a truncated cone with a base diameter of \(8 \mathrm{ft},\) a top diameter of \(2 \mathrm{ft}\), and a height of \(100 \mathrm{ft}\). Determine the resultant vertical force that the ropes must resist when the completely submerged tree is cut. The specific gravity of the wood is approximately 0.6.

Step by step solution

01

Calculate densities

First, calculate the density of the water and wood. Standard values can be used for this. The density of water \(\rho = 1000 kg/m^3\) and acceleration due to gravity \(g = 9.81 m/s^2\). The density of the tree is given by \(\rho_{tree} = 0.6 \times \rho = 0.6 \times 1000 = 600 kg/m^3\). The specific gravity of the wood is 0.6.

02

Compute the differential equations

The differential equations can now be formed. For the buoyant force on an infinitesimally small slice of the tree at height z, dFb = \(\rho g (π r^2 dz)\), where r is the radius at height z, which linearly scales with the height. Simultaneously, the weight of the same slice of the tree is dFw = \(\rho_{tree} g (π r^2 dz)\). Taking the limit from 0 to H, where H is the height of the tree, the integral can be taken to find the resultant force.

03

Solve the integral

Finally, the integral \(F = \int_{0}^{H} (F_b - F_w) dz\) needs to be solved. This will give the resultant upward force, F, on the tree. This integral may require the application of integration techniques such as substitution or by parts.

04

Convert units

Since the problem was given in imperial units whilst the calculations were made in metric, the final step is to convert the unit of the resultant force from Newtons to pounds by using the conversion relation 1 lbf = 4.44822 N. The resultant force is the force that needs to be resisted by the ropes.

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