A wooden block floating in seawater has two thirds of its volume submerged. When the block is placed in mineral oil, 10% of its volume is submerged.
(
a) Find the density of the wooden block. (The density of seawater is 1.024 g/cm3.)
(b) Find the density of the mineral oil.
To find the density of the wooden block, we can use the principle of buoyancy. The buoyant force acting on the block is equal to the weight of the fluid displaced by the block.
(a) Let's denote the volume of the wooden block as V and its density as D. Since two-thirds of its volume is submerged in seawater, the volume of water displaced by the block is (2/3)V.
The weight of the water displaced is equal to the buoyant force, which can be calculated using the volume of water displaced and the density of seawater:
Buoyant force = Weight of water displaced = Volume of water displaced * Density of seawater
Buoyant force = (2/3)V * 1.024 g/cm^3
The weight of the wooden block is equal to its volume multiplied by its density:
Weight of wooden block = V * D
Since the block is floating, the weight of water displaced is equal to the weight of the block:
Weight of wooden block = Buoyant force
V * D = (2/3)V * 1.024 g/cm^3
Simplifying the equation, we get:
D = (2/3) * 1.024 g/cm^3
D = 0.6827 g/cm^3
Therefore, the density of the wooden block is 0.6827 g/cm^3.
(b) To find the density of the mineral oil using the same principle, we can use the volume submerged in mineral oil, which is 10% of the total volume (0.1V).
The weight of the mineral oil displaced is equal to the buoyant force, which can be calculated using the volume of oil displaced and the density of the oil:
Buoyant force = Weight of oil displaced = Volume of oil displaced * Density of oil
Buoyant force = (0.1V) * Density of oil
Since the weight of the wooden block is equal to the buoyant force, we have:
Weight of oil displaced = Weight of wooden block
(0.1V) * Density of oil = V * 0.6827 g/cm^3
Simplifying the equation, we get:
Density of oil = (0.6827 g/cm^3) / 0.1
Density of oil = 6.827 g/cm^3
Therefore, the density of the mineral oil is 6.827 g/cm^3.
To find the density of the wooden block, we can use the principle of buoyancy. The buoyant force acting on an object submerged in a fluid is equal to the weight of the fluid displaced by the object. The buoyant force can be calculated using the equation:
Buoyant force = density of fluid * volume of fluid displaced * gravitational acceleration
Let's denote the density of the wooden block as "ρ", and the volume of the wooden block as "V".
(a) To find the density of the wooden block:
1. In seawater:
- Two thirds of the volume of the wooden block is submerged, so the volume of fluid displaced is (2/3)V.
- The density of seawater is given as 1.024 g/cm^3.
- The gravitational acceleration is approximately 9.8 m/s^2.
Using the equation for buoyant force, we have:
Buoyant force = (1.024 g/cm^3) * ((2/3)V) * (9.8 m/s^2)
2. In mineral oil:
- 10% of the volume of the wooden block is submerged, so the volume of fluid displaced is 0.1V.
- Let's denote the density of the mineral oil as "ρ_m".
Using the equation for buoyant force, we have:
Buoyant force = ρ_m * (0.1V) * (9.8 m/s^2)
Since the density of the wooden block is the same in both cases, we can equate the two expressions for buoyant force:
(1.024 g/cm^3) * ((2/3)V) * (9.8 m/s^2) = ρ_m * (0.1V) * (9.8 m/s^2)
Simplifying the equation:
ρ * (2/3) = ρ_m * 0.1
Dividing both sides by (2/3):
ρ = ρ_m * 0.1 * (3/2)
ρ = 0.15 * ρ_m
The density of the wooden block is 0.15 times the density of the mineral oil.
(b) To find the density of the mineral oil:
We can use the same equation from part (a) and rearrange it to solve for ρ_m:
ρ_m = ρ / 0.15
Substituting the known density of seawater into the equation:
ρ_m = 1.024 g/cm^3 / 0.15
Calculating:
ρ_m ≈ 6.8267 g/cm^3
Therefore, the density of the mineral oil is approximately 6.83 g/cm^3.