Force calculations on a hemisphere

In summary, the conversation discusses finding the force on a hemisphere submerged in water at a certain height. The use of spherical coordinates and integration is suggested, but it is mentioned that using Archimedes' principle would be a simpler approach. The importance of considering cylindrical symmetry and the vertical components of the force elements is also mentioned.
  • #1
stevemir
1
0
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!
 
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  • #2
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).
 
  • #3


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!
 

Related to Force calculations on a hemisphere

1. What is the formula for calculating the force on a hemisphere?

The formula for calculating the force on a hemisphere is F = (2/3) * π * r^2 * p, where F is the force, π is pi (approximately 3.14), r is the radius of the hemisphere, and p is the pressure.

2. How does the force on a hemisphere change with pressure?

The force on a hemisphere is directly proportional to the pressure applied. This means that as the pressure increases, the force also increases, and vice versa.

3. What is the significance of the radius in force calculations on a hemisphere?

The radius of the hemisphere is a crucial factor in force calculations because it determines the surface area of the hemisphere. The larger the radius, the greater the surface area and therefore the greater the force.

4. Can force calculations on a hemisphere be used for other shapes?

Yes, the formula for force calculations on a hemisphere can be modified and used for other shapes as long as they have a curved surface. However, the radius and shape of the object will affect the accuracy of the calculation.

5. How is the force on a hemisphere affected by the material it is made of?

The material of the hemisphere can affect the force calculation in two ways. Firstly, different materials have different densities, which can impact the pressure applied. Secondly, the strength and rigidity of the material can affect how it responds to pressure, which can impact the force calculation.

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