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Assume that an object has negligible height, i.e., height is approximately $0m$.

If we immerse this object in water, will it float or sink? In order to find this out, we need to calculate the net force acting on the object. I am considering $3$ main forces acting on the object. The first of these forces is the force exerted by the molecules of the liquid at the top of the object in the downward direction, which we will call $F_d$. The second force is the weight force acting on the object in the downward direction, which is $F_W$. Thirdly, we have the force exerted by the molecules of the liquid at the bottom of the object in the upward direction, which we will call $F_u$.

To find the net force acting on the object: $$F_{net} = (F_d + F_W) - F_u$$

The pressure exerted by the liquid at the top of the object is $P_t$ and that exerted at the bottom of the object is $P_b$. As the height is approximately $0$, we will consider the $P_t$ and $P_b$ to be equal. Also, again as the height is negligible, the area at the top and the bottom of the object, i.e., $A_t$ and $A_b$ will also be equal.

We know that $P=\frac{F}{A}$. Thus to find force, $F = PA$.

$F_d = P_tA_t$

$F_u = P_dA_d$

$P_tA_t = P_dA_d$ as pressure and area at the top and bottom are equal.

$$F_{net} = (P_tA_t + F_W) - P_dA_d$$ $$F_{net} = (PA + F_W) - PA$$ $$F_{net} = F_W$$

According to this, the net force acting on the object at any height would always be $F_W$ and, of course, always in the downward direction, thus signifying that it would always sink, regardless of density of the substance, height in the liquid, volume of liquid displaced, etc.

Is this conclusion true or is my understanding of the concept or the equations flawed at any point?

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Your conclusion is correct. Since the height is zero and $F_W$ is a (finite) constant, the density is infinite. An object sinks when its density is higher than the density of the fluid. An object with infinite density therefore always has higher density than the surrounding fluid and always sinks.

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    $\begingroup$ It may be worth spelling out explicitly that to avoid the unphysical infinite density you should take into account that for real objects short objects will tend to be lighter and so $F_W$ will also tend to zero as the height tends to zero $\endgroup$
    – By Symmetry
    Commented Aug 11, 2023 at 15:24
  • $\begingroup$ Thanks, I added the fact that the weight is constant as an explicit part of the reasoning. $\endgroup$
    – Dale
    Commented Aug 11, 2023 at 15:31
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for an object whose thickness is vanishingly small, its volume is vanishingly small too and as its thickness trends to zero so does its volume. With no volume, it displaces no liquid and therefore no buoyant force is generated, which means it cannot float.

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  • $\begingroup$ but thin films will float in cause of the tension of the surface. $\endgroup$
    – trula
    Commented Aug 11, 2023 at 16:41
  • $\begingroup$ @trula, surface tension was not an issue in the original question. $\endgroup$
    – niels nielsen
    Commented Aug 11, 2023 at 17:03

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