a modified U-tube: the right arm is shorter than the left arm. The open end of the right arm is height d = 10.0 cm above the laboratory bench. The radius throughout the tube is 1.60 cm. Water is gradually poured into the open end of the left arm until the water begins to flow out the open end of the right arm. Then a liquid of density 0.770 g/cm3 is gradually added to the left arm until its height in that arm is 6.70 cm (it does not mix with the water). How much water (in cm3) flows out of the right arm?
To find the volume of water that flows out of the right arm, we need to consider the pressure difference between the two ends of the U-tube.
At equilibrium, the pressure at any given level must be the same throughout the tube. So, we can equate the pressures at the level where the water exits the right arm and the level where the liquid of density 0.770 g/cm³ ends in the left arm.
Let's break down the problem step by step:
1. Calculate the pressure at the exit of the right arm:
The pressure at a given depth within a fluid can be calculated using the formula P = ρgh, where P is the pressure, ρ is the density, g is the acceleration due to gravity, and h is the depth.
The pressure at the exit of the right arm is given by P_right = ρ_water * g * h_right, where ρ_water represents the density of water and h_right is the height of the water column in the right arm.
2. Calculate the pressure at the end of the liquid in the left arm:
The pressure at this level consists of two components: the pressure due to the liquid and the pressure due to the height difference between the liquid and the water in the right arm.
The pressure at the end of the liquid in the left arm is given by P_left = (ρ_liquid * g * h_liquid) + (ρ_water * g * (h_liquid - h_right)), where ρ_liquid represents the density of the liquid added and h_liquid is the height of the liquid column in the left arm.
3. Set the pressures at the two locations equal to each other and solve for the height h_right:
ρ_water * g * h_right = (ρ_liquid * g * h_liquid) + (ρ_water * g * (h_liquid - h_right))
Simplifying, we get: h_right = (ρ_liquid * h_liquid) / (ρ_liquid + ρ_water)
4. Calculate the volume of water that flows out of the right arm:
Given that the radius throughout the tube is 1.60 cm, we can use the formula for the volume of a cylindrical tube: V = π * r² * h, where V is the volume, r is the radius, and h is the height.
The volume of water that flows out of the right arm can be calculated using V_water = π * (1.60 cm)² * h_right.
Plug in the given values: ρ_water = 1.0 g/cm³, ρ_liquid = 0.770 g/cm³, h_liquid = 6.70 cm in the equations above to find the height h_right and then calculate the volume V_water.
By following these steps, you can find the answer to the question.