A cube with side length 1.53m of solid Styrofoam is completely submerged in fres
ID: 2123056 • Letter: A
Question
A cube with side length 1.53m of solid Styrofoam is completely submerged in fresh water such that the top of the Styrofoam block i parallel to the water surface and is 2.55m beneath the surface. The density of Styrofoam varies but you have determined that this block has a density of 105kg/m^3 and also know the density of fresh water is 1000kg/m^3
a) what is the buoyant force acting upon the Styrofoam block?
b) what is the total pressure on the top of the block due to water pressure?
c) what is the total pressure on the bottom of the block due to water pressure?
d)what is the total force acting on the top of the block (due to water pressure)?
e)what is the total force acting on the bottom of the block (due to water pressure)?
f)what is the total force acting on the block due to water pressure only?
Please explain your answers. I would like to learn... Thanks
Explanation / Answer
The sides are identical in area, and have the same depth distribution, therefore they also have the same pressure distribution, and consequently the same total force resulting from hydrostatic pressure, exerted perpendicular to the plane of the surface of each side.
There are two pairs of opposing sides, therefore the resultant horizontal forces balance in both orthogonal directions, and the resultant force is zero.
The upward force on the cube is the pressure on the bottom surface integrated over its area. The surface is at constant depth, so the pressure is constant. Therefore, the integral of the pressure over the area of the horizontal bottom surface of the cube is the hydrostatic pressure at that depth multiplied by the area of the bottom surface.
Similarly, the downward force on the cube is the pressure on the top surface integrated over its area. The surface is at constant depth, so the pressure is constant. Therefore, the integral of the pressure over the area of the horizontal top surface of the cube is the hydrostatic pressure at that depth multiplied by the area of the top surface.
As this is a cube, the top and bottom surfaces are identical in shape and area, and the pressure difference between the top and bottom of the cube is directly proportional to the depth difference, and the resultant force difference is exactly equal to the weight of the fluid that would occupy the volume of the cube in its absence.
This means that the resultant upward force on the cube is equal to the weight of the fluid that would fit into the volume of the cube, and the downward force on the cube is its weight, in the absence of external forces.
a) buoyant force=volume water displaced(vol. of cube)*density water*gravity=35.13 kN
b ) total pressure top= height below water surface*density water*gravity(9.81)=2.55*1000*9.81=25 kPa
c) total pressure bottom=height bottom surface below water*density water*gravity(9.81)=40.02 kPa
d) force =pressure *top surface area of cube=25*1.53*1.53=58.52 kN
e) force = pressure*bottom surface area of cube =40.02 *1.53*1.53=93.68kN
f ) total vertical force=93.68-58.52=35.16kN