A cube of ice whose edges measure 20.0 mm is floating in a glass of ice-cold water, and one of t

A cube of ice whose edges measure 20.0 mm is floating in a...

A cube of ice whose edges measure 20.0 $\mathrm{mm}$ is floating in a glass of ice-cold water, and one of the ice cube's faces is parallel to the water's surface. (a) How far below the water surface is the bottom face of the ice cube? (b) Ice-cold ethyl alcohol is gently poured onto the water surface to form a layer 5.00 $\mathrm{mm}$ thick above the water. The alcohol does not mix with the water. When the ice cube again attains hydrostatic equilibrium, what is the distance from the top of the water to the bottom face of the block? (c) Additional cold ethyl alcohol is poured onto the water's surface until the top surface of the alcohol coincides with the top surface of the ice cube (in hydrostatic equilibrium). How thick is the required layer of ethyl alcohol?

Key Concepts

Hydrostatic Equilibrium

Hydrostatic equilibrium refers to the state where an object floating in a fluid is in balance because its weight is exactly balanced by the buoyant force. This concept is used to determine the submerged depth of the object and to analyze how additional layers of different fluids, such as immiscible liquids, affect the equilibrium condition.

Density and Its Role in Buoyancy

The density of a fluid and the object determines the fraction of the volume submerged when the object floats. Specifically, comparing the density of the object with that of the fluid allows for calculating the equilibrium position where the weight of the displaced fluid equals the object's weight, crucial for solving buoyancy problems.

Stratification of Immiscible Fluids

When two immiscible fluids are present, each with different densities, they form distinct layers. Understanding the stratification and the interfaces between these layers is important for solving problems where objects interact with more than one fluid, as the buoyant force must be calculated by considering the contributions from each layer.

Archimedes' Principle

This principle states that any body wholly or partially immersed in a fluid experiences an upward force equal in magnitude to the weight of the fluid displaced by the body. It is fundamental to understanding why and how objects float or sink in a fluid, and it is used to establish equilibrium conditions of buoyant objects.

Buoyant Force

The buoyant force is the upward force exerted by a fluid on an object immersed in it, which originates from the pressure differences at different depths. Its magnitude, as given by Archimedes' Principle, is essential for analyzing the floating or sinking behavior of objects and predicting the portion of the object that remains submerged.

Transcript

00:01 So here we've got a block of ice floating, a cube of ice floating in some water.

00:07 And it's perfectly floating so that the top face is parallel with the surface of the water.

00:13 And the first part is how far below the water's surface is the bottom face of the block.

00:18 So tell me that length.

00:22 So we know the density of water and the density of ice.

00:27 And so with those two gibbons, we can determine.

00:31 That that ratio of the densities is going to be how much is below the surface.

00:36 So the density of water is, you know, 91 .7 % of the way there compared to water, and therefore the amount underneath, i'm going to call this y.

00:49 Y is equal to 91 .7 % of the total 20mm side length.

00:59 So we get that the amount that it's submerged is 18 .34 millimeters.

01:09 Part b is asking if we added some ethyl alcohol, some ice cold, ethyl alcohol is poured into the water surface.

01:18 They say ice cold so that we don't have to worry about melting.

01:21 Poured into the water surface to form a layer of 5 millimeters thick above the water.

01:27 So now we've added some alcohol.

01:30 Above the water.

01:35 So once we've added that ethyl alcohol to the top, we've got a new situation.

01:41 And again, they want us to find the new y, the new distance from the top of the water to the base of the cube.

01:47 So i'm going to draw a free body diagram for this one because this gets a little complex for the ice cube.

01:53 So i know that i've got my mass of the ice cube times gravity down.

01:58 But in this case, i've got some i've got two buoyant forces.

02:06 I've got the buoyant force from the alcohol and the buoyant force from the water.

02:14 And so i can, because it's in hydrostatic equilibrium, it's not moving.

02:18 Once it reaches equilibrium, i can set these equal to each other, and i can get the buoyant force from the water plus the point force from the alcohol will be equal to the mass times g up the cube.

02:33 And so here i can replace the formula for point force.

02:38 So density of the water comes the volume of the cube.

02:42 Again, the volume of the cube will say upper, meaning the volume that is in the, oh, sorry, lower, the volume that is in the water, yeah, times g plus the, this is for the point force from the alcohol, which is the density of the alcohol.

03:04 Times the volume of the upper section times g, and then that equal the mass of the cube times g...

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