A circular disc having a diameter of 125cm is immersed in water so that the distance of its perimeter measured vertically below the water surface varies between 60cm and 150cm. Find the total force due to the water acting on one side of the lamina, and the vertical distance of the centre of pressure below the surface.
Can someone please show me how to solve this problem?
center of mass = ((150-60)/2)+60 = 1.05
area = (1/4)*pi*D^2 = 1.227
Force = density*g*area*centerOfMass
Force = 1*9.81*1.227*1.05
Force = 12.639KN
can show the step ? pls
The pressure at a depth of x is p(x)=ρgx.
The total force acting on any immersed surface can be calculated by integrating over the depth, i.e. x=h1 to h2.
h1 and h2 in this particular case is 60 and 150 cm respectively.
To find the centre of pressure, integrate x*p(x) over h1 to h2, and divide by the total pressure obtained above.
do you have any idea from where this quesiton was taken please? Like a book ?
How did you get 1.05 as your center of mass?
Based on your procedure, center of mass is supposed to be 105
Ok
To solve this problem, we need to apply the concept of hydrostatic pressure and understand the relationships between the force, pressure, and centroid of a submerged object.
Let's start by finding the total force due to the water acting on one side of the disc.
1. Find the area of the circular disc:
The area of a circle is given by the formula A = πr^2, where r is the radius.
In this case, the diameter of the disc is given as 125 cm, so the radius (r) is half of the diameter, i.e., 125 cm / 2 = 62.5 cm.
Therefore, the area of the circular disc is A = π(62.5 cm)^2.
2. Find the average vertical distance of the perimeter below the water surface:
In this case, the distance of the perimeter varies between 60 cm and 150 cm.
The average vertical distance is the average of these two distances, which is (60 cm + 150 cm) / 2 = 105 cm.
3. Calculate the pressure at the average vertical distance below the water surface:
The pressure (P) at a certain depth in a fluid is given by the equation P = ρgh, where ρ is the density of the fluid, g is the acceleration due to gravity, and h is the depth.
The density of water is approximately 1000 kg/m^3, and we can convert the average vertical distance to meters by dividing by 100, resulting in 1.05 m.
Thus, the pressure at the average vertical distance below the water surface is P = 1000 kg/m^3 * 9.8 m/s^2 * 1.05 m.
4. Find the force acting on the submerged part of the disc:
The force (F) acting on the submerged part of the disc is given by the equation F = P * A, where P is the pressure and A is the area.
Substituting the values we calculated, F = (1000 kg/m^3 * 9.8 m/s^2 * 1.05 m) * (π * (62.5 cm)^2).
To find the vertical distance of the center of pressure below the water surface, we need to know the moment of the force.
5. Calculate the moment of the force:
The moment of a force is given by the equation M = F * h, where F is the force and h is the perpendicular distance between the line of action of the force and the axis of rotation.
In this case, the axis of rotation is the center of the circular disc, and the perpendicular distance is the vertical distance from the center of the circular disc to the center of pressure.
Let's denote the vertical distance from the center of the disc to the center of pressure as x.
The moment of the force is then M = F * x.
6. Set up the equation to find x:
To calculate x, we can consider the equilibrium condition, where the sum of the clockwise moments is equal to the sum of the counterclockwise moments.
In this case, the clockwise moment is M and the counterclockwise moment is due to the weight of the disc acting at the center of the disc. The weight of the disc is given by the equation W = m * g, where m is the mass of the disc and g is the acceleration due to gravity.
The mass of the disc can be calculated using the density (ρ) and the volume (V) of the disc. The volume of the disc is given by the equation V = A * h, where A is the area of the disc and h is the thickness of the disc. Let's denote the density as ρ, the thickness as t, and the acceleration due to gravity as g.
The counterclockwise moment is then M_counterclockwise = W * x_counterclockwise, where W = ρ * V * g.
Setting up the equation, we have F * x = ρ * A * t * g * x.
7. Solve the equation for x:
Rearranging the equation, x = (ρ * A * t * g * x) / F.
Substituting the known values, we have x = (ρ * π * (62.5 cm)^2 * t * g * x) / F.
Solving for x, we find x = (ρ * π * (62.5 cm)^2 * t * g) / F.
By following these steps and applying the appropriate formulas, you should be able to solve the problem and find the total force due to the water acting on one side of the disc and the vertical distance of the center of pressure below the water surface.