Solving Fluid Statics Problem: Centre of Pressure

[AppleBite]

Ok, I've come across this problem in fluid statics, but seem to be getting the integration wrong:

"A semicircular plane is submerged vertically in a homogeneous liquid with its diameter d at the free surface. At what depth s is the centre of pressure?"

The answer should be

s = (3*pi*d) / 32

Any ideas?

 

[FredGarvin]

Forget the integration and try the form of the solution found in equation 11-7b in the following link:

http://books.google.com/books?id=Km...X&oi=book_result&resnum=2&ct=result#PPA488,M1

 

[oswaler]

The solution to this problem involves using the principles of fluid statics and the concept of the centre of pressure. The centre of pressure is the point at which the total force acting on a submerged surface is equal to the weight of the fluid above that surface. In this case, the semicircular plane is submerged vertically, so we can use the formula for the centre of pressure on a vertical surface:

s = (3*pi*d) / 32

where s is the depth of the centre of pressure, d is the diameter of the semicircular plane, and pi is the mathematical constant pi.

To understand this formula, we need to first understand the concept of the centroid of a semicircle. The centroid is the point at which the area of a semicircle is evenly distributed. In this case, the centroid of the semicircular plane is located at a distance of d/4 from the base of the semicircle.

Now, let's consider the forces acting on the semicircular plane. The weight of the fluid above the plane is equal to the weight of the fluid displaced by the plane. This can be calculated using the formula for the volume of a cylinder (since the semicircular plane is essentially a half-cylinder):

V = (pi*d^2*s) / 8

where V is the volume of the fluid displaced, d is the diameter of the semicircular plane, and s is the depth of the centre of pressure. So, the weight of the fluid above the plane is:

F = ρgV = (pi*d^2*s*ρg) / 8

where ρ is the density of the fluid and g is the acceleration due to gravity.

Now, the total force acting on the plane is equal to the weight of the fluid above the plane plus the weight of the plane itself. The weight of the plane can be calculated using its surface area and the density of the material it is made of:

F' = ρgA = (pi*d^2*ρg) / 4

where A is the surface area of the semicircular plane.

To find the depth of the centre of pressure, we need to set these two forces equal to each other and solve for s:

F = F'

(pi*d^2*s*ρg) / 8 = (pi*d^2*ρg) / 4

s = (3*pi*d) /