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blizzard12345
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Homework Statement
A circular cylinder, of cross sectional radius r = (2 + A) m, f
oats in equilibrium on the
horizontal
flat surface of a liquid. The horizontal plane through the axis of the cylinder is
parallel to the fl
at liquid surface, which is very large in extent. A cartesian Oxyz coordinate
system is constructed such that the axis of the cylinder is aligned with the y coordinate axis and
the z coordinate axis is vertically upwards. The origin O is located on the axis of the cylinder.
The region above the liquid surface is filled with air at atmospheric pressure patm = (110) kPa.
The density of the material making up the cylinder is dc = (4)x10^2 kg/m3
and
the density of the liquid is dl = (30)x10^2 kg/m3
The only body force is gravity, which is constant and acts in the negative z direction. The acceleration due to
gravity is g = 9:81 m/s^2
. You should assume that the cylinder is long and that you may neglect end effects.
If the interface between the air and the
liquid is determined by the plane z = r cos (beta), determine
the value of (beta) in degrees and give an indication of the numerical accuracy of your result.
Homework Equations
The Attempt at a Solution
i have attempted this a few different ways which involve solving the final equation with the bisection method. i have taken the centroid of the cylinder where z = 0, i have then used the equal pressure equations for the points of the bottom of the cylinder directly below the centroid and a place on the surface of the water, the following equation is what i believe to be correct however the answer given tell me i may have gone wrong (since altering the density of the materials does not change the angle found to what i think sufficiently)
f_value = (dl*r*g - dl*g*(r*cosd(x))) * ( (pi*r^2/2) + (pi/2 * (r^2 - ((r*sind(x))^2))) - (pi*r^2*dc*g))
my question is have i gone about this correctly or is there another way I am not seeing?
it may take me a little while to get back i work part time on weekend so may struggle to find time to reply thanks for any help given :)