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I came across this question where, In a container there's water at the bottom , kerosene on top and an ice cube floating between them and I was asked to calculate the ratio of height of cube in ice to that in water:

enter image description here

Now till now my intuition for up thrust was that there needs to be some fluid below the object to give an upward perpendicular force.As simple as this:

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Therefore for this question I thought there should be no upthrust from kerosene but I was surprised when I saw the free body diagram in the solution:

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And therefore now I wish to understand:

  1. What's wrong with my understanding of upthrust?
  2. Where is the upthrust from kerosene actually coming from and How does upthrust acts on a body in general?
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  • $\begingroup$ Are you able to do a derivation for the buoyant force when there is only water with an ice cube floating in it? $\endgroup$ Commented Oct 14, 2021 at 16:44
  • $\begingroup$ @DavidWhite , yes I am. $\endgroup$
    – Kashi
    Commented Oct 15, 2021 at 15:48

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This diagram was used in an answer to this question Why does a fluid push upward on a body fully or partially submerged in it?

enter image description here

The spring is first at its natural length, then released, showing that there is an up-thrust on the piston due to the height difference (and weight) of the liquid.

In a similar way, even though there is no Kerosene underneath the cube, the weight of it (transmitted through the water) adds to the upward pressure and force pressing on the underneath the cube. On the top side of the cube, there is only the weight of a reduced depth of Kerosene pressing down.

So the result is that there is also an up-thrust due to the weight of the Kerosene displaced, as well as due to the weight of water displaced, as described by Archimede's principle.

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  • $\begingroup$ I'm not able to understand how that diagram and its explanation is related to this specific problem, that (diagram) is simply a concept of pressure due to height difference which can simply be explained by $${p}_{final}={p}_{atm}+\rho g{h}_{object}$$....can't it? Secondly, ** the weight of it (transmitted through the water) adds to the upward pressure ** , from my understanding pressure and upthrust don't depend on mass , $$Upthrust={V}_{FluidDisplaced}\times {\rho }_{Fluid}\times g$$ ..... $\endgroup$
    – Kashi
    Commented Oct 15, 2021 at 15:42
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A buoyant force results when the pressure pushing up from the bottom of an object is greater than the pressure pushing down from above. For your situation, there is an increase in the pressure as you go down from the top of the cube through the kerosene and another increase as you go down to the bottom through the water.

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  • $\begingroup$ I understand why water will give an upthrust to the cube, but not why the kerosene will do so, there is no surface for kerosene to apply an upward perpendicular force. $\endgroup$
    – Kashi
    Commented Oct 15, 2021 at 15:47
  • $\begingroup$ The weight of the kerosene increases the pressure in the water. $\endgroup$
    – R.W. Bird
    Commented Oct 15, 2021 at 17:47
  • $\begingroup$ Can you please explain "weight of kerosene increases pressure in the water" this part more clearly. Maybe with a diagram...Its just hard to understand how exactly this thing is happening? $\endgroup$
    – Kashi
    Commented Oct 16, 2021 at 14:30
  • $\begingroup$ You have atmospheric pressure at the surface of the kerosene. This increases as you go down through the kerosene. It increases some more as you go down through the water. The extra pressure in the water at the bottom of the cube contributed by the kerosene is the same as the pressure at the bottom of the kerosene. $\endgroup$
    – R.W. Bird
    Commented Oct 17, 2021 at 13:29
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For water only with an ice cube floating in it, the buoyant force occurs due to the the difference in force between the bottom of the ice cube and the top of the ice cube. This difference in force is due to the pressure that accrues due to the depth at the bottom of the ice cube, via the equation $P=\rho g h$, where $h$ is the liquid depth at the bottom of the ice cube.

When the ice cube is floating in water that has a kerosene layer on top of it, it sinks in the kerosene as shown in the posted diagrams. To analyze this situation, it is helpful to make an assumption. For the sake of the derivation, assume that the ice cube is floating with an orientation such that its top and bottom surfaces are horizontal, ensuring that every part of the top surface experiences the same pressure, and every part of the bottom surface experiences the same (higher) pressure.

The following variable definitions apply:

$A_T$ = the area of the top of the ice cube

$A_B$ = the area of the bottom of the ice cube

$h_1$ = the kerosene depth at the top of the ice cube

$h_2$ = the depth of kerosene at the bottom of the kerosene layer

$h_3$ = the depth of water (not kerosene) at the bottom of the ice cube

$P_T$ = the pressure on the top surface of the ice cube

$F_T$ = the force on the top of the ice cube

$P_B$ = the pressure on the bottom of the ice cube

$F_B$ = the force on the bottom of the ice cube

$\rho_k$ = the density of kerosene

$\rho_w$ = the density of water

From these definitions, it is seen that:

$P_T = \rho_k g h_1$

$F_T = \rho_k g h_1 A_T$

$P_B = \rho_k g h_2 + \rho_w g h_3$

$F_B = [\rho_k g h_2 + \rho_w g h_3]A_B$

Assuming that the area of the top of the ice cube equals the area of the bottom of the ice cube, $A_T = A_B = A$. The buoyant force on the ice cube is the difference between the force on the bottom of the ice cube and the force on the top of the ice cube. Thus,

$F_{buoyant} = F_B - F_T = \rho_kgh_2A+ \rho_wgh_3A - \rho_kgh_1A$

$F_{buoyant} = \rho_kg(h_2 - h_1)A + \rho_wgh_3A$

The term $\rho_kg(h_2-h_1)A$ is seen to be the weight of the kerosene that is displaced by the ice cube. The term $\rho_wgh_3A$ is seen to be the weight of the water that is displaced by the ice cube. This means that the buoyant force on the ice cube is the sum of the weight of kerosene and the weight of water that is displaced by the ice cube.

An alternative viewpoint may make this solution more intuitive. With no kerosene layer, the force on the bottom of the ice cube is due to water only. With a kerosene layer, the weight of the kerosene increases the pressure throughout the water column, so the force from the water on the bottom of the ice cube is higher than expected, meaning that the kerosene definitely has an influence on the pressure and force on the bottom of the ice cube, even though the bottom of the ice cube is not in kerosene.

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For a prismatic body like that, it is enough to calculate the pressure difference between the bottom surface and top surface. The force is this difference times the area of the surface (where area of top and bottom surfaces are supposed equal).

$$F = A(p_b - p_t) = A(\mu_q gh_1 + \mu_q gh_2 + \mu_w gh_3 - \mu_q gh_1 = A(\mu_q gh_2 + \mu_w gh_3)$$ where
$\mu_q$ is the specific mass of querosene
$\mu_w$ is the specific mass of water
$h_1$ is the height from the querose free surface to the top surface of the object
$h_2$ is the height from the liquid interface to the top surface of the object.
$h_3$ is the height from the liquid surface to the bottom surface of the object.

As can be seen, the result is exactly the sum of the displaced weight of querosene and water.

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