Density of a Bar of Soap/ buoyancy

[raisatantuico]

Homework Statement

A rectangular bar of soap floats with 3.5 cm extending below the water surface and 1.5 cm above. what is its density?

Homework Equations

p(rho)=m/v

The Attempt at a Solution

I don't know how to compute the volume since we are given only two lengths.
Athough, i think a better way to approach this problem will be with buoyancy.

To start,can we consider this block to be in equilibrium since it is floating? I am confused because it is also partially submerged.

 

[Stonebridge]

Yes. This question is solved using the idea of buoyancy and Archimedes Principle.
Here's a way of starting...
Let the area of the top and bottom surface be A m²
The volume of the soap is therefore 0.05A m³ [0.05 is the height/thickness of the soap]

Now because the soap is floating, its weight is exactly balanced by the upthrust due to the weight of the volume of water it is displacing.
Weight = mass x g
Mass = volume x density

Does this help you to get started?

 

[raisatantuico]

how did you get .05A?

So if it is floating, mg=Buoyancy.
mg=density x volume x g

how do i get the mass if i don't know the density?
Sorry, there are two unknowns still.. density and mass. i don't know how to get either with just the volume given

 

[Stonebridge]

If there is 3.5cm of soap below water and 1.5 above, then the thickness of the soap is 3.5 + 1.5 = 5cm
5cm = 0.05m

Volume is area of cross section times length/thickness
If the area of cross section (of the top and bottom of the soap) is A then the volume of the soap is 0.05A

When the soap is floating, only 3.5cm (0.035m) is below the water.
This means that the volume under water is 0.035A
The volume of water displaced is V=0.035A
The mass of water is volume V times density of water ρ_w
The weight of water is mass times g
=ρ_w x 0.035A x g
This equals the buoyancy force.

This force must be equal to the weight of the soap.
The weight of the soap is its volume (0.05A) times its density.
If you equate the buoyancy force to the weight of the soap you will get a value for the density of the soap.
A and g cancel out. You need a value for the density of water.

 

[rwisz]

Yes, we can consider the bar of soap to be in equilibrium since it is floating and not sinking or rising in the water. To calculate its density, we can use the equation p=m/v, where p is density, m is mass, and v is volume. Since we do not have the mass of the soap bar, we can use the concept of buoyancy to calculate its density.

Buoyancy is the upward force exerted by a fluid on an object that is partially or fully submerged in it. This force is equal to the weight of the fluid displaced by the object. In this case, the fluid is water and the object is the soap bar.

We can use the following equation to calculate the buoyant force on the soap bar:

Fb = ρVg

Where Fb is the buoyant force, ρ is the density of water (1000 kg/m³), V is the volume of the soap bar, and g is the acceleration due to gravity (9.8 m/s²).

Since the soap bar is in equilibrium, the buoyant force must be equal to the weight of the soap bar. We can calculate the weight of the soap bar using the formula:

Fg = mg

Where Fg is the weight, m is the mass, and g is the acceleration due to gravity.

Since the soap bar is floating, the buoyant force is equal to the weight of the soap bar. Therefore, we can equate the two equations:

Fb = Fg

ρVg = mg

We can rearrange this equation to solve for the volume of the soap bar:

V = m/ρg

Now, we need to find the mass of the soap bar. We can use the fact that the soap bar is partially submerged to calculate its volume. The volume of the submerged part of the soap bar can be calculated as follows:

Vsubmerged = A*h

Where A is the cross-sectional area of the soap bar and h is the depth to which it is submerged.

The total volume of the soap bar can be calculated as:

Vtotal = l*w*t

Where l is the length, w is the width, and t is the thickness of the soap bar.

Since we know the length and width of the soap bar, we can calculate its cross-sectional area as:

A = l*w

Now, we can substitute these values into the previous equation to find the volume of the soap bar:

Vtotal