A solid cylinder (radius = 0.170 m, height = 0.150 m) has a mass of 7.00 kg. This cylinder is floating in water. Then oil (ñ = 0.866 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?
What situation is shown in the drawing?
Application of Archimedes Principle should yield the answer
To determine how much of the height of the cylinder is in the oil, we need to calculate the heights of both the water and the oil.
Step 1: Calculate the height of the water.
The given density of water (ñ_w) is 1000 kg/m^3.
To find the height of the water, we'll use Archimedes' principle: the buoyant force on the cylinder is equal to the weight of the water displaced by the cylinder.
Buoyant force = Weight of water displaced by the cylinder
Buoyant force = Weight of the cylinder [Since the cylinder is floating]
Weight of water displaced by the cylinder = ñ_w * Volume of the cylinder * g
= ñ_w * (π * r^2 * h_w) * g [Volume of cylinder = π * r^2 * height of water (h_w)]
= 1000 kg/m^3 * (π * (0.170 m)^2 * h_w) * 9.8 m/s^2 [g = 9.8 m/s^2]
= 1665.542 N [Approximately]
The weight of the cylinder can be calculated as follows:
Weight of the cylinder = Mass of the cylinder * g
= 7.00 kg * 9.8 m/s^2
= 68.60 N [Approximately]
Therefore, the height of the water can be calculated as follows:
Buoyant force = Weight of water displaced by the cylinder
1665.542 N = 68.60 N
(1000 kg/m^3 * π * (0.170 m)^2 * h_w * 9.8 m/s^2) - (7.00 kg * 9.8 m/s^2) = 0
Now, we can solve this equation to find the height of the water (h_w).
Step 2: Calculate the height of the oil.
The given density of the oil (ñ_o) is 0.866 kg/m^3.
To find the height of the oil, we'll use the same principle as before: the buoyant force on the cylinder is equal to the weight of the oil displaced by the cylinder.
Buoyant force = Weight of oil displaced by the cylinder
Buoyant force = Weight of the cylinder [Since the cylinder is floating]
Weight of oil displaced by the cylinder = ñ_o * Volume of the cylinder * g
= ñ_o * (π * r^2 * h_o) * g [Volume of cylinder = π * r^2 * height of oil (h_o)]
= 0.866 kg/m^3 * (π * (0.170 m)^2 * h_o) * 9.8 m/s^2 [g = 9.8 m/s^2]
= 47.029 N [Approximately]
The weight of the cylinder (already calculated) can be considered as the weight of the oil displaced by the cylinder.
Therefore, the height of the oil can be calculated as follows:
Buoyant force = Weight of oil displaced by the cylinder
47.029 N = 68.60 N
(0.866 kg/m^3 * π * (0.170 m)^2 * h_o * 9.8 m/s^2) - (7.00 kg * 9.8 m/s^2) = 0
Now, we can solve this equation to find the height of the oil (h_o).
Step 3: Calculate the height of the cylinder in the oil.
To find the height of the cylinder in the oil, we subtract the height of the water from the total height of the cylinder.
Height of the cylinder in the oil = total height of the cylinder - height of the water
Finally, we have the answer: the height of the cylinder in the oil.
To find out how much of the height of the cylinder is in the oil, we need to consider the densities of the cylinder, water, and oil.
Let's first calculate the density of the cylinder. The formula for the density (ρ) is mass (m) divided by volume (V), so:
ρ = m / V
Given that the mass (m) of the cylinder is 7.00 kg and the cylinder is a solid shape, the volume (V) can be calculated using the formula for the volume of a cylinder:
V = πr^2h
where π is a constant (approximately equal to 3.14159), r is the radius of the cylinder (0.170 m), and h is the height of the cylinder (0.150 m).
Substituting the values into the volume formula, we get:
V = (3.14159)(0.170^2)(0.150)
Now we can calculate the density of the cylinder using the mass (7.00 kg) and the calculated volume (V).
Next, let's calculate the density of water. The density of water is usually given as 1000 kg/m^3.
Now, let's determine the height of the part of the cylinder submerged in water. Since the cylinder is floating, the buoyant force acting on the cylinder is equal to the weight of the water displaced by the submerged part of the cylinder.
The buoyant force is given by the formula:
F_b = ρ_water * V_submerged * g
where ρ_water is the density of water (1000 kg/m^3), V_submerged is the volume of the submerged part of the cylinder, and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Solving for V_submerged, we get:
V_submerged = F_b / (ρ_water * g)
Finally, we can calculate the height of the part of the cylinder submerged in water using the formula for the volume of a cylinder:
V_submerged = πr^2 * h_submerged
where r is the radius of the cylinder (0.170 m) and h_submerged is the height of the submerged part of the cylinder.
Substituting the values into the equation, we can solve for h_submerged.
The remaining height of the cylinder, which is in the oil, is equal to the total height of the cylinder minus the height submerged in water.
Hope this helps!