A solid cylinder (radius = 0.150 m, height = 0.120 m) has a mass of 7.00 kg. This cylinder is floating in water. Then oil ( = 725 kg/m3) is poured on top of the water until the situation shown in the drawing results. How much of the height of the cylinder is in the oil?
To determine how much of the height of the cylinder is in the oil, we need to find the height of the oil layer.
Let's start by finding the volume of the cylinder, which is given by the formula:
Volume = π * radius^2 * height
Given that radius = 0.150 m and height = 0.120 m, we can calculate the volume of the cylinder:
Volume = π * (0.150 m)^2 * 0.120 m
Volume ≈ 0.00851 m^3
Since the cylinder is floating in water, the volume of the water displaced by the cylinder is equal to the volume of the cylinder:
Volume of water = 0.00851 m^3
Now, we need to find the volume of oil. We can do this by subtracting the volume of water displaced by the cylinder from the total volume of the cylinder:
Volume of oil = Total volume of the cylinder - Volume of water
Volume of oil = π * (0.150 m)^2 * 0.120 m - 0.00851 m^3
Volume of oil ≈ 0.0145 m^3
Next, we can calculate the height of the oil layer by dividing the volume of oil by the cross-sectional area:
Height of oil = Volume of oil / (π * radius^2)
Height of oil = 0.0145 m^3 / (π * (0.150 m)^2)
Height of oil ≈ 0.216 m
So, approximately 0.216 m of the height of the cylinder is in the oil.
To find out how much of the height of the cylinder is in the oil, we need to determine the equilibrium position of the cylinder in the water-oil mixture.
Step 1: Calculate the volume of the cylinder:
The volume of a cylinder is given by the formula V = πr^2h, where r is the radius and h is the height.
V = π(0.150 m)^2 * 0.120 m
V ≈ 0.00849 m^3
Step 2: Calculate the buoyant force acting on the cylinder:
The buoyant force on the cylinder is equal to the weight of the fluid it displaces. Since the cylinder is floating, the buoyant force is equal to the weight of the cylinder.
Buoyant force = weight of cylinder = mass of cylinder * acceleration due to gravity
Buoyant force = 7.00 kg * 9.8 m/s^2
Buoyant force ≈ 68.6 N
Step 3: Determine the weight of the water displaced by the cylinder:
Using Archimedes' principle, the weight of the water displaced by the cylinder is equal to the buoyant force.
Weight of water displaced = buoyant force
Weight of water displaced ≈ 68.6 N
Step 4: Calculate the volume of water displaced by the cylinder:
The volume of water displaced is equal to the volume of the cylinder that is submerged in water. Since the density of water is 1000 kg/m^3, we can use the formula:
Volume of water displaced = Weight of water displaced / density of water
Volume of water displaced ≈ 68.6 N / (1000 kg/m^3 * 9.8 m/s^2)
Volume of water displaced ≈ 0.00702 m^3
Step 5: Calculate the volume of oil on top of the water:
The volume of oil can be obtained by subtracting the volume of water displaced from the total volume of the cylinder:
Volume of oil = Volume of cylinder - Volume of water displaced
Volume of oil ≈ 0.00849 m^3 - 0.00702 m^3
Volume of oil ≈ 0.00147 m^3
Step 6: Calculate the height of the oil in the cylinder:
Since the volume of the oil is equal to the cross-sectional area of the cylinder multiplied by the height of the oil,
Volume of oil = πr^2 * height of oil
0.00147 m^3 = π(0.150 m)^2 * height of oil
height of oil ≈ 0.00984 m
Therefore, approximately 0.00984 m, or 9.84 cm, of the height of the cylinder is in the oil.