Answer to Question #261616 in Mechanical Engineering for Jansen

Question

A wooden log of 0.8 m diameter and 6 m length is floating in a river water. Find the depth of wooden log in water when the sp. gr. Of the wooden log is 0.7.

Answer 1

$$mg=\rho_wgV_\text{submerged},\\\space\\ V_\text{submerged}=\frac{m}{\rho_w}=\frac{SG·\rho_w·V}{\rho_w},\\\space\\ V_\text{submerged}=SG·V=SG·\frac{\pi d^2}{4}L.$$

The volume above the water surface is

$$\Delta V=V-V_\text{submerged}=(1-SG)·\frac{\pi d^2}{4}L.$$

Divide by length to find the area of the segment above the water:

$$A=(1-SG)·\frac{\pi d^2}{4}.$$

The area of the segment can be expressed in terms of areas of a triangle and sector:

$$A=A_s-A_t=\pi\theta-x·(r-h),\\ \theta=2\arctan\frac xh.\\\space\\ A=2\pi\arctan\frac xh-x·(r-h).\\\space\\ x^2=r^2-(r-h)^2=2rh-h^2.\\\space\\ h=0.26\text{ m}.$$

Therefore, the depth is

$$D=d-h=0.54\text{ m}.$$