# What density of an object of more than one substance affects the buoyant force?

If we immerse a sealed cup of a liquid in a fluid, what density is taken in consideration when it comes to the buoyant force, the density of the cup itself (the substance in contact with the fluid) or the whole cup (with the liquid inside)?

Also, I was reading in a textbook about how buoyant force affects weighing using balances, and I went over a problem where the density of the weighing bottle was ignored and only the density of the substance inside was taken in mind to calculate the correction for the buoyant force.

So, is that okay to just ignore the density of the bottle itself (even though its mass is way greater than the substance inside)?

• Buoyant force depends on the volume of body and the density of fluid it is immersed into. Regarding bottle, I think the book has considered it's mass to be negligible than what it contains. But it would be better if you could provide the snapshot of the text book. Commented Jan 4, 2021 at 16:47
• What do you mean by "a liquid in a fluid"? Commented Jan 4, 2021 at 16:54
• @BobD a cup of a liquid, in a fluid Commented Jan 4, 2021 at 17:12
• @SteelCubes The book is about analytical chemistry and the part I mentioned talks about an equation to correct the error resulting from buoyant forces when using electronic balances. The book says "The same buoyant force acts on the container during both weighings", meaning before and after we put the substance inside the bottle. Commented Jan 4, 2021 at 17:17

Usually, by buoyant force, people mean the upward Archimede's force equal to the weight of the same volume of the body filled by the surrounding liquid. It is the global effect of the surface force due to a gradient of pressure in the surrounding liquid.

Therefore, the substance the body is made of and the container's weight do not enter in the buoyant force expression. They do enter the body's weight, which is the force that has to be compared with Archimede's force to establish the buoyancy of the body.

• "Therefore, the substance the body is made of and the container's weight do not enter in the buoyant force expression". Hmm. The total weight of the container plus contents will determine the amount of fluid displaced. So unless the weight of the container is negligible relative to the contents, shouldn't it be taken into account to determine the buoyant force? Commented Jan 4, 2021 at 17:41
• @BobD as I wrote at the beginning, the usual definition (see for example en.wikipedia.org/wiki/Buoyancy ) of the buoyant force is the upward force exerted by a fluid that opposes the weight of a partially or fully immersed object. Your concern would be justified if by buoyant force one defines the algebraic sum of Archimede's force and weight. With the definition I am using the volume of the body determines the buoyant force, independently of the weight of the body. It is evident that the volume of the body in the present case is the volume of the container filled with the liquid Commented Jan 4, 2021 at 17:52
• "With the definition I am using the volume of the body determines the buoyant force". Still don't follow you. The buoyant force is the volume of the fluid displaced by the body. If the density of the body is equal to or greater than the density of the fluid, then the volume of the body determines the buoyant force. Commented Jan 4, 2021 at 17:57
• @BobD I cannot understand what is not convincing to you. It seems that you agree that "The buoyant force is the (weight of a) volume of the fluid displaced by the body (i.e. a volume $V$ of the fluid equal to the volume $V$ of the body). I also agree that once we have the total mass ($M$) of the body, the comparison of the mass density of the fluid and $M/V$ determines the resulting force. I suspect we use a different definition of buoyant force. Commented Jan 4, 2021 at 18:27
• The volume $V$ of the displaced fluid only equals the volume $V$ of the body if the density of the body is equal to or greater the density of the liquid. And the density of the body depends on the total mass divided by the volume. But I choose not to argue this any further since we obviously are having a failure to communicate, no offense. Commented Jan 4, 2021 at 18:35

If we immerse a sealed cup of a liquid in a fluid, what density is taken in consideration when it comes to the buoyant force, the density of the cup itself (the substance in contact with the fluid) or the whole cup (with the liquid inside)?

Although it should be noted that in many buoyancy examples the mass of the container itself is neglected, technically you need to account for both the container and its contents in order to determine the volume of the fluid displaced, which, in turn, determines the buoyant force.

In order to determine the upward buoyant force, you need to determine the volume of the fluid displaced by the cup+liquid. To do that, you need to determine the density of the combination of the cup+liquid. To calculate the density, take the sum of the mass of the cup and the mass of the liquid and divide by the overall volume of the cup+liquid. That gives you the density of the combination of the cup+liquid.

If that density is equal to or greater than the density of the fluid in which it is immersed, it will either float completely submerged or sink. Either way, the upward buoyant force will be the same and will equal the weight of the fluid having volume equal to the volume of the cup+liquid.

On the other hand, if that density is less than the density of the fluid, the cup+liquid will float partially submerged. The submerged volume of the object will equal the volume of the fluid displaced, $$V_f$$, or

$$V_{f}=V_{cup+liq}\frac{ρ_{cup+liq}}{ρ_{f}}$$

Where $$\rho _f$$ is the density of the fluid. The upward buoyant force will then be

$$F_{buoyant}=\rho _{f}V_{f}g$$

Hope this helps.

Neither one nor the other of the two densities are taken, but rather both. For example, say the bottle had a mass of 5 g and volume (plastic only) of 10cm3, the bottle would have a density of 0.5g/cm3. And say the liquid inside had a mass of 100g and volume of 100cm3, the liquid would have a density of 1g/cm3. However, when combined, they have a density of 0.95454... g/cm3 (105g/110cm3). However, as you can see, this is extremely close to the density of the liquid (1g/cm3), which is why the bottle density is often ignored, as it has a negligible effect upon the density.

Technically, however, it is this total average density that should be used to calculate buoyant force.