[Physics] Question about buoyancy

Question

I took an empty clear sphere with radius r, put some weight inside this sphere, and put in into the water. I calculate the buoyancy force as:

Volume of the sphere = Volume of the water displaced ($V_{\text{displaced}}$) = $(4/3) \pi r^3$.

Weight of the displaced water = Buoyancy force ($B$) = $V_{\text{displaced}} \times$ density of water.

$$W_{\text{total}}= W_{\text{sphere}} + W_{\text{weight}}.$$

Then, the sphere will float when $B > W_{\text{total}}$, otherwise it will sink. Is this correct? So the air inside the sphere will affect sphere's ability to float. My confusion is that I assume that I have already considered this effect during the aforementioned calculations (since density of the sphere itself is low). Is that right? Please note that the sphere is not flexible.

Answer 1

Then, the sphere will float when B>Wtotal, otherwise it will sink. Is this correct? So the air inside the sphere will affect sphere's ability to float.

Yes but it will be a small effect. The density of air is nearly a thousand times less than that of water.

My confusion is that I assume that I have already considered this effect during the aforementioned calculations

The big question becomes how are you measuring your masses, both of the sphere, the objects you place in the sphere and the density of the water you are putting your sphere in?

Your scales are sitting in air, so they don't measure the actual weight of the object they measure the difference between the object's weight and the weight of the air displaced by the object.

So if you weigh everything, water, sphere and objects using normal scales in normal air you are implicitly taking account of the mass of the air by pretending everything (including the water) is slightly less dense than it really is.