A rectangular block of wood floats in water with two-third of its volume immersed. When placed in another liquid, it floats with half of its volume immersed. Calculate the relative density of the liquid.
2/3 * Dw = 1/2 * D
relative density= D/Dw=4/3
To calculate the relative density of the liquid, we need to compare the densities of the liquid and the wood.
Let's assume the density of the wood is ρw and the density of the liquid is ρl.
In the first scenario, when two-thirds of the wood's volume is immersed in water, we can use Archimedes' principle to determine the relationship between the density of the wood and the density of water.
According to Archimedes' principle, the buoyant force acting on an object submerged in a fluid is equal to the weight of the fluid displaced by the object. For a floating object, the buoyant force is equal to the weight of the object.
The weight of the wood = (Volume of wood submerged in water) * (density of water) * (gravity)
Given that two-thirds of the volume is submerged, the volume of the wood submerged in water is (2/3) * (volume of the wood).
The weight of the wood = (2/3) * (volume of the wood) * (density of water) * (gravity)
On the other hand, the weight of the wood can also be expressed as (volume of the wood) * (density of wood) * (gravity).
Equating the two expressions for the weight of the wood, we can find the relationship between the density of the wood (ρw) and the density of water (ρwater):
(2/3) * (ρw) * (volume of the wood) * (gravity) = (volume of the wood) * (ρwater) * (gravity)
Simplifying the equation, we get:
ρw = (2/3) * ρwater
Now let's move on to the second scenario when the wood is placed in another liquid and half of its volume is immersed.
Following the same reasoning, the weight of the wood can be expressed as:
(1/2) * (volume of the wood) * (ρl) * (gravity)
Thus, the relationship between the density of the wood (ρw) and the density of the second liquid (ρl) becomes:
ρw = (1/2) * ρl
To calculate the relative density of the liquid, we can divide the density of the second liquid by the density of water, as both densities are related to the density of the wood:
(relative density of the liquid) = (ρl) / (ρwater)
Substituting the expressions we found for ρw in terms of ρwater and ρl, we get:
(relative density of the liquid) = (1/2) * ρl / (2/3) * ρwater
Simplifying further, we get:
(relative density of the liquid) = (3/4) * (ρl / ρwater)
Hence, the relative density of the liquid is (3/4) times the ratio of the density of the second liquid to the density of water.