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. (5
Db = density of the block.
Dw = density of water.
Dl = density of the other liquid.
Db/Dw * Vb = 2Vb/3.
Db/Dw = 2/3.
Dw/Db = 3/2 = 1.5.
Dw = 1.5Db.
Db/Dl = 1/2.
Dl = 2Db.
Dl/Dw = 2Db/1.5Db = 4/3 = 1.333 = Relative density of the liquid.
To determine the relative density of the liquid, we need to use the concept of buoyancy.
Let's assume the volume of the rectangular block of wood is V and its density is ρw.
Given that two-thirds of the volume is immersed in water, it means the buoyant force equals the weight of the water displaced by the immersed volume.
The buoyant force buoyant_wo for the wood block in water can be calculated using Archimedes' principle:
buoyant_wo = ρw * g * (2/3)V
where g is the acceleration due to gravity.
Now, when the wood block is placed in another liquid, we know it floats with half of its volume immersed. Again, the buoyant force buoyant_liq for the wood block in this liquid can be calculated using Archimedes' principle:
buoyant_liq = ρliq * g * (1/2)V
where ρliq is the density of the liquid.
Knowing that the buoyant forces in both cases are equal (the block floats without sinking or rising), we can set up an equation:
ρw * g * (2/3)V = ρliq * g * (1/2)V
Simplifying the equation and canceling out the common terms:
(2/3)ρw = (1/2)ρliq
To find the relative density ρrel of the liquid, we can rearrange the equation:
ρrel = ρliq / ρw = (3/2) * (2/3)
Finally, simplifying the equation gives us:
ρrel = 1
Therefore, the relative density of the liquid is 1.
To answer this question, we need to understand the concept of relative density and how it is related to the immersion of an object in a liquid.
Relative density, also known as specific gravity, is the ratio of the density of a substance to the density of a reference substance, usually water. It is a dimensionless quantity that indicates how dense (heavier or lighter) a substance is compared to water.
Now, let's consider the situation given in the problem: when the rectangular block of wood is placed in water, two-thirds of its volume is immersed. This means that the density of the wooden block is two-thirds that of the water.
To calculate the relative density of the liquid, we need to know the density of the wooden block and the density of the liquid. Let's assume the density of the wooden block is Dw and the density of the liquid is Dl.
In the first case, when the wooden block floats in water with two-thirds of its volume immersed, we can use the concept of buoyancy. The buoyant force acting on the block is equal to the weight of the water displaced. Since two-thirds of the block's volume is immersed, the weight of the displaced water is two-thirds of the weight of the block. Mathematically, this can be expressed as:
Buoyant force = weight of water displaced
(Dw x Volume of the block) x g = (Dw - Dw/2) x (Volume of the block) x g
Simplifying this equation, we find:
(Dw x Volume of the block) = (Dw - Dw/2) x (Volume of the block)
(Dw x Volume of the block) = (Dw/2) x (Volume of the block)
We can cancel out the volume of the block on both sides, resulting in:
Dw = Dw/2
Simplifying further:
2 = 1
This is not possible, which indicates that there is an error in the problem statement or the given information.
Therefore, we cannot determine the relative density of the liquid based on the provided information.