Average Volume of Half a Sphere

[Icheb]

A sphere is submerged half way in water and pulled out at a constant speed. Now I have to find out the average volume of the part of the sphere that is under water.

The volume formula would be $$4/3 \pi r^3$$, now the part of the sphere that is under water at first would be $$2/3 \pi r^3$$.

Initially I thought I could assume the sphere to be a cube and then calculate the volume of the cube under water. Half of that volume would be the result then since the velocity is constant. However I'm not sure if that approximation is alright?


Edit: Never mind, found it! :)

 

[berkeman]

Glad you found it. Did you use integration to get the answer, or is there a simpler way?

 

[m2003]

I would approach this question by first clarifying some assumptions and parameters. For example, is the sphere perfectly spherical or is it slightly distorted? Is the water level exactly halfway up the sphere or is it slightly higher or lower? These factors can affect the calculations and results.

Assuming that the sphere is perfectly spherical and the water level is exactly halfway up, then your initial approach of approximating the sphere as a cube and finding the volume of the cube under water is a valid method. This is because the volume of a cube is simply length x width x height, which is the same as the formula for volume of a sphere under water (2/3 \pi r^3). However, it is important to note that this approximation may not be accurate for larger or more distorted spheres.

Another approach would be to use the formula for the volume of a spherical cap, which is the volume of a spherical section cut by a plane. In this case, the plane would be the water level and the volume of the spherical cap would represent the volume of the sphere under water. This formula is 1/3 \pi h^2 (3r - h), where h is the height of the spherical cap and r is the radius of the sphere. This method may be more accurate for larger or distorted spheres.

In conclusion, both the cube approximation and the spherical cap formula can be used to find the average volume of half a sphere submerged in water. It is important to consider the assumptions and limitations of each method and choose the most appropriate one for the specific situation.