Question

Click Reset and drop object A back into the water. About what percentage of object A is under the water? How is this percentage related to the density of object A?

Answer 1

Archimedes' principle allows finding the percentage of material submerged in the fluid and the general relationship that for a body to float is:   Body density must be less than fluid density

Archimedes' principle says that the force of a fluid on a body is equal to the weight of the liquid dislodged

               B = ρ_{fluid} g V_{fluid}

Where B is the thrust, rho and V the density and volume of the fluid, respectively, and g is the acceleration of gravity (g = 9.80 m / s²).

In the attachment we can see a diagram of a body in water, let's apply the equilibrium condition

              Σ F = 0

              B - W = 0

              W = m g

Density is a very useful concept that relates the mass and volume of a body

              ρ = m / V

              m = ρ V

we substitute

             B = ρ_{body} g V_ {body}

In the attachment we see that part of the body is below the fluid, so the volume of fluid dislodged is

              V_ {fluid} = l w h_ {under}

Where V is the volume of fluid dislodged, l and w are the length and width of the body, and h_{under}  is the distance of the submerged body.

We substitute

             ρ_ {fluid} g l w h_ {under} = ρ_ {body} g l w h_ {total}

             $\frac{\rho_{body}}{\rho_{fluid} } = \frac{h_{under}}{h_{total}}$          (1)

Let's analyze this expression that gives the ratio of the densities to the ratio of the height of the submerged body.

In the table we give some densities and the height of a body

Material      density        total height    % under

                   ρ(kg/m³)            (m)

water (fluid)    1.00 10³          1.00

Ice                  0.917 10³         0.10            91.7%

Oak                0.710 10³         0.10            71%

Pine                0.373 10³        0.10            37.3%

Aluminum       2.70 10³          0.10          100%

Polyethylene  0.94 10³         0.94           94%

Let's calculate the submerged height, where we will use water as a fluid, let's use equation 1

                 h_ {under}  =  $\frac{\rho_{body}}{\rho_{fluid}} \ h_{total}$

Fluid: Water    

Material: Ice

           h under = $\frac{0.917}{1.00} \ 1.00$  

           h under = 0.917 m

To calculate the percentage of the sunken body we use

           % under = h_ {under} / h_ {total}

           % under = $\frac{0.917}{1.00}$   100

           % under = 91.7%

Material: Pine

             h under = $\frac{0.373}{0.10} \ 0.10$

             h under = 0.0373 m

             % under = $\frac{0.0373}{0.10}$   100

             % under = 37.3%

Material: Aluminum

            In this case all the matrix is ​​submerged

            %under = 100%

The calculated values ​​are also shown in the last column of the table.

It is observed that for a material to float in another, its density must be less than the density of the fluid, in the case of a ship, the volume of the ship is very large, therefore its apparent density is less than that of water.

In conclusion, using Archimedes' principle we can find the percentage of bound material   and the general generation that for a body to float is:

 material floats in another its density must be less than the density of the fluid

Learn more about Archimedes' Principle here:

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