Force calculations on a hemisphere

[stevemir]

I need to find the force on a hemisphere below a certain height H in water. The hemisphere is resting at the bottom. The radius of the hsphere is r. I think i need to use dF = dPdA and use spherical coordinates to integrate but do not know how to form the triple integral required!

 

[Doc Al]

Doing the integration is one way to solve it, but not the easiest. Make use of Archimedes' principle to find the buoyant force on a submerged half-ball.

But if you'd like to do the integral, be sure to take advantage of cylindrical symmetry. (I assume the axis is vertical.) And realize that only the vertical components of the force elements remain (the horizontal components cancel).

 

[thepatientmental]

To calculate the force on a hemisphere below a certain height H in water, you can use the formula dF = dPdA, where dP is the pressure at a certain point on the hemisphere and dA is the area at that point.

To set up the triple integral, you will need to use spherical coordinates. In spherical coordinates, the radius r is constant, the angle θ ranges from 0 to π/2 (since we are only considering the bottom half of the hemisphere), and the angle φ ranges from 0 to 2π.

The pressure at a certain point on the hemisphere can be calculated using the hydrostatic pressure formula: dP = ρgh, where ρ is the density of water, g is the acceleration due to gravity, and h is the height of the water above that point.

To find the area at a certain point on the hemisphere, you can use the formula dA = r^2sinθdθdφ.

Putting all of this together, the triple integral for the force on the hemisphere would be:

∫∫∫ dF = ∫∫∫ ρgh * r^2sinθdθdφ

Where the limits of integration for θ are from 0 to π/2 and the limits for φ are from 0 to 2π.

Once you have evaluated this integral, you will have the total force on the hemisphere below the height H in water. Keep in mind that this force will be a vector, so you will need to consider both the magnitude and direction of the force.

I hope this helps you in your calculations. Good luck!