A solid when weighed in air has a weight W. It's weight in water is W¹ and when finally weighed in a liquid the weight is W²find the relative density of the liquid in terms of W,
W¹ and W²
The weight of a solid in air is equal to its actual weight (W) minus the buoyant force exerted on it by the air. This buoyant force can be calculated using Archimedes' principle, which states that the buoyant force is equal to the weight of the fluid displaced by the solid. Thus, the weight of the solid in air (W) is given by:
W = weight of solid - buoyant force in air
The weight of the solid in water (W¹) is equal to its actual weight (W) minus the buoyant force exerted on it by the water. Similarly, the weight of the solid in the liquid (W²) is given by:
W² = weight of solid - buoyant force in liquid
Since both water and the liquid are fluids, the buoyant force is calculated the same way for both cases, using Archimedes' principle.
Now, let's find the relative density of the liquid in terms of W, W¹, and W². Relative density is defined as the ratio of the density of the liquid to the density of water. Let's denote the densities as ρ₁ for water and ρ₂ for the liquid.
To find the relative density, we need to compare the buoyant forces in water and the liquid. Since the buoyant force is equal to the weight of the fluid displaced, the buoyant force in water is equal to the weight of the water displaced by the solid, and the buoyant force in the liquid is equal to the weight of the liquid displaced by the solid.
The weight of the water displaced is given by ρ₁ * volume of the solid. Similarly, the weight of the liquid displaced is given by ρ₂ * volume of the solid.
Therefore, we have:
W¹ = W - ρ₁ * volume of the solid
W² = W - ρ₂ * volume of the solid
Subtracting these two equations gives:
W¹ - W² = ρ₁ * volume of the solid - ρ₂ * volume of the solid
W¹ - W² = (ρ₁ - ρ₂) * volume of the solid
Dividing both sides by the volume of the solid gives:
(W¹ - W²) / volume of the solid = (ρ₁ - ρ₂)
Finally, since W = weight of the solid and weight is equal to the mass multiplied by the acceleration due to gravity, we have:
W = ρ * volume of the solid * g
where ρ is the density of the solid. Rearranging this equation gives:
ρ = W / (volume of the solid * g)
Substituting this equation into the previous equation gives:
(W¹ - W²) / volume of the solid = (W / (volume of the solid * g)) - ρ₂
Cancelling out the volume of the solid on both sides gives:
W¹ - W² = W / g - ρ₂ * volume of the solid
Now, dividing both sides by the volume of the solid gives:
(W¹ - W²) / volume of the solid = (W / (volume of the solid * g)) - ρ₂
Cancelling out the volume of the solid on both sides gives:
(W¹ - W²) / volume of the solid = W / (volume of the solid * g) - ρ₂
Finally, rearranging the equation gives:
ρ₂ = W / (volume of the solid * g) - (W¹ - W²) / volume of the solid
Therefore, the relative density of the liquid in terms of W, W¹, and W² is:
ρ₂ = W / (volume of the solid * g) - (W¹ - W²) / volume of the solid
To find the relative density of the liquid in terms of W, W¹, and W², we need to understand the principle of buoyancy.
The weight of an object in air (W) is equal to the weight of the object in water (W¹) plus the buoyant force acting on the object, which is equal to the weight of the displaced water.
In equation form, we have:
W = W¹ + Buoyant force
Similarly, the weight of the object in the liquid (W²) is equal to the weight of the object in air (W) plus the buoyant force acting on the object, which is equal to the weight of the displaced liquid.
In equation form, we have:
W² = W + Buoyant force
Now, let's find the relative density (ρ) of the liquid.
Relative density is defined as the ratio of the density of a substance to the density of a reference substance (usually water).
For the object in water:
W = W¹ + Buoyant force
Buoyant force = W - W¹
For the object in liquid:
W² = W + Buoyant force
Buoyant force = W² - W
Since the buoyant force is the weight of the displaced substance, we can equate the two expressions for the buoyant force:
W - W¹ = W² - W
Simplifying the equation, we get:
2W - W¹ = W²
Now, let's solve for the relative density of the liquid (ρ):
Relative density (ρ) = Density of the liquid / Density of water
Since density is mass (m) divided by volume (V), we can rewrite the equation as:
ρ = (m/V)liquid / (m/V)water
Now, let's substitute mass and volume with their respective weights:
ρ = (W²/V)liquid / (W¹/V)water
Since the volume (V) of the object is the same in both cases, it cancels out:
ρ = (W²/W)liquid / (W¹/W)water
ρ = (W²/W)liquid * (W/W¹)water
Finally, substituting W - W¹ = W² - W, we have:
ρ = (W²/W)liquid * (W/W¹)water
Therefore, the relative density of the liquid in terms of W, W¹, and W² is (W²/W)liquid * (W/W¹)water.