![what density floats in water what density floats in water](https://4.bp.blogspot.com/-_0pHf4YAffw/WnGYHSrR9PI/AAAAAAAAA04/9gWP6D_9ltYvCBzP4xYXff3f5QUtY0yZwCLcBGAs/s1600/Density%2Bdefinition.png)
It's not too difficult to solve, but I get a value for a of 0.6189* R. I will skip the details - you can do this for a homework problem if you like. If I set this cap volume to 0.228 the volume of the full sphere, then I can solve for a. This says that the volume of the cap (the top part) would be: It's clear it will be larger than the radius of the planet, but by how much? Instead of deriving the formula for the volume of a partial sphere - I will use this Wikipedia page for a spherical cap. You can see I need to find the value for h which is the depth the planet would go underwater. How deep would this be? Here is a picture. Using my density of Saturn, 77.2% of it would be underwater. This means that the volume of water displaced will be the volume of Saturn multiplied by the ratio of the densities. But how much? If I call the volume of the planet underwater V d (d is for displacement), then I can write: This means that only part of the planet would be underwater. If the planet could float (see below), how deep would the water need to be? For a floating object, the buoyancy force is equal to the gravitational force. Also, I will assume that in this region of water, the gravitational field is constant and pointing straight down since the planet is so large. How much water would you need for Saturn to float? Let's assume for now that this is some ginormous planet with as much water as we need. So it seems logical that Saturn would also float. Things with a density less than water float - things like ducks, tiny rocks and gravy. If you want a more detailed derivation of the buoyancy force - check out this post about the Magdeburg Water Bridge. And here you see that if the density of an object is less than the density of water, that object will float. In order to be at equilibrium, the object would be just partially submerged.
![what density floats in water what density floats in water](https://ssl.c.photoshelter.com/img-get/I0000ijdN4xNjKaA/s/860/860/Fphoto-67039001A-6CC.jpg)
So, if the density of water is greater than the density of the object, the buoyancy force when the object is fully submerged will be greater than the weight.
![what density floats in water what density floats in water](https://thumbs.dreamstime.com/z/ilustraci%C3%B3n-vectorial-de-densidad-relativa-r%C3%A9gimen-flotaci%C3%B3n-o-hundimiento-etiquetado-materiales-etiquetados-explicaci%C3%B3n-la-165034642.jpg)
The only thing that is different is the density. This would be written as:īoth the weight and the buoyancy force have the same V o g term. For the buoyancy force, I can calculate this as the weight of the water displaced. Here I just wrote the mass of the object as the product of the density of the object (ρ o) and the volume of the object ( V o).