A piece of wood is floating at the surface of some water as illustrated (the diagram is not to scale). The wood has a circular cross section and a height h = 4.0 cm. The density of water is 1.00 g/cm3. The distance y from the surface of the water to the bottom of the wood is 3.0 cm. The density of the wood is (in g/cm3):
Question 1 options:
A)Impossible to determine because the area of the cross section is not given.
B)12
C)0.75
D)0.25
the volume underwater is area*h=area*3
mass of water displced=densitywater*volume
= 1g/cm^3*area*3
This is equal to the weight of the wood:
weight wood=densitywood*volume
= densitywood(area*4cm)
set them equal
1*area*3=densitywood*area*4
density wood=.75g/cm^3
To determine the density of the wood, 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.
In this case, the wood is floating at the surface of the water, so the buoyant force is equal to the weight of the wood. The weight of an object can be calculated using the equation:
Weight = density × volume × gravity
Given that the height (h) of the wood is 4.0 cm and the distance from the surface of the water to the bottom of the wood (y) is 3.0 cm, we can calculate the volume of the wood using the formula for the volume of a cylinder:
Volume = π × radius^2 × height
Since the wood has a circular cross section, we need to find the radius of the wood. We can use the Pythagorean theorem to find the radius:
radius^2 = (height/2)^2 + y^2
Simplifying this equation gives us:
radius^2 = (4.0 cm / 2)^2 + (3.0 cm)^2
radius^2 = (2.0 cm)^2 + (3.0 cm)^2
radius^2 = 4.0 cm^2 + 9.0 cm^2
radius^2 = 13.0 cm^2
Taking the square root of both sides gives us:
radius = √13.0 cm
Now that we have the radius, we can calculate the volume of the wood:
Volume = π × (√13.0 cm)^2 × 4.0 cm
Volume = π × 13.0 cm × 4.0 cm
Volume ≈ 163.3628179 cm^3
Next, we need to calculate the weight of the wood using the density of water. The weight can be calculated as:
Weight = density × volume × gravity
Since the wood is floating, the weight is equal to the buoyant force, which is given by the weight of the fluid displaced by the wood. Therefore:
Weight = density_water × volume × gravity
Given that the density of water is 1.00 g/cm^3 and the volume is approximately 163.3628179 cm^3, we can calculate the weight:
Weight ≈ 1.00 g/cm^3 × 163.3628179 cm^3 × 9.8 m/s^2
Weight ≈ 1605.2235342 g
Now we have the weight of the wood. To calculate the density of the wood, we need to divide the weight by the volume of the wood:
Density = Weight / Volume
Density ≈ 1605.2235342 g / 163.3628179 cm^3
Density ≈ 9.8246453 g/cm^3
Therefore, the density of the wood is approximately 9.82 g/cm^3, which is not listed in the answer choices. However, based on the given options, the correct answer would be A) Impossible to determine because the area of the cross section is not given.
To determine the density of the wood, we need to consider the principle of buoyancy. According to Archimedes' principle, the buoyant force acting on an object partially or fully submerged in a fluid is equal to the weight of the fluid displaced by the object.
In this scenario, the wood is floating at the surface of the water, which means it is in equilibrium. The weight of the wood is balanced by the buoyant force acting on it. The weight of the wood can be calculated by multiplying its volume by its density, and the buoyant force can be determined by multiplying the volume of water displaced by the wood by the density of water.
Let's assume the density of the wood is represented by ρ (rho). The volume of the wood can be calculated by multiplying the cross-sectional area of the wood (A) by its height (h). However, given that the area of the cross-section is not given, we cannot directly determine the volume and therefore the density of the wood.
Hence, the answer to Question 1 is:
A) Impossible to determine because the area of the cross section is not given.