Problem 81 - Fundamentals of Physics

Question

Shows a modified U-tube: the right arm is shorter than the left arm. The open end of the right arm is height \(d=10.0 \mathrm{~cm}\) above the laboratory bench. The radius throughout the tube is \(1.50 \mathrm{~cm}\). 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 \(0.80 \mathrm{~g} / \mathrm{cm}^{3}\) is gradually added to the left arm until its height in that arm is \(8.0 \mathrm{~cm}\) (it does not mix with the water). How much water flows out of the right arm?

Step-by-Step Solution

Answer

6.4 cm of water flowed out from right arm.

Step 1: Understand the Problem

We're dealing with a U-tube with different substances on each side. The left arm is initially filled with water, causing it to flow out of the right arm. Then, a liquid is added to the left arm, and we need to find out how much water flows out as a result.

Step 2: Analyze the Initial Water Setup

Initially, water is added until it starts flowing out of the right arm. Since the open end of the right arm is 10.0 cm above the bench, the column of water in the left arm balances at this height before any liquid is added. At this point, water in the right arm has reached 10.0 cm.

Step 3: Calculate Initial Pressure Balance

The pressure in the water on both sides can be described by the hydrostatic pressure equation, \[ P = \rho g h \]Given the left arm's water column height establishes equilibrium with the right arm having 10.0 cm water height.

Step 4: Introduce New Liquid and Determine New Heights

When the 0.80 g/cm³ liquid is added to a height of 8.0 cm in the left arm, it will exert its own pressure, pushing some of the water in the right arm out. The pressure changes, so new heights need calculating.

Step 5: Calculate Combined Effect Height

Let the height of the water remaining in the left arm be denoted as \( h_w \). The pressure from the liquid is \[ P = \rho_l g \times 8.0 \ \text{cm} \]. The total pressure on the left side equals the pressure of the remaining water and the liquid: \[ \rho_l g \times 8.0 + \rho_w g h_w = \rho_w g \times 10.0 \].

Step 6: Solve for Remaining Water Height on Left

Replacing densities (water density \( \rho_w = 1.0 \ \text{g/cm}^3 \) and liquid density \( \rho_l = 0.80 \ \text{g/cm}^3 \)), solve for \( h_w \) using \[ 0.80 \times 8.0 + 1.0 \times h_w = 1.0 \times 10.0 \].Solving gives \( h_w = 3.6 \ \text{cm} \).

Step 7: Calculate Water Flowed Out

Since the initial height of water was 10.0 cm and the remaining water height after adding the liquid is 3.6 cm, the height difference corresponds to the amount of water that flowed out: \[ 10.0 - 3.6 = 6.4 \ \text{cm} \]. Convert this to volume using the cross-sectional area: \[ V = \pi r^2 h = \pi \times (1.5 \ \text{cm})^2 \times 6.4 \].

Key Concepts

U-tube experiments

U-tube experiments are interesting demonstrations of fluid dynamics principles. In these setups, we use a U-shaped tube filled with various liquids to explore concepts like hydrostatic pressure and equilibrium. In this particular problem, a U-tube has one shorter arm where water flows out as another liquid is added.
A crucial aspect of U-tube experiments is understanding how different heights and densities of liquids affect the flow between the arms. In our scenario, water is initially poured into the left arm until it reaches a balance with the right arm. Next, another liquid is added without mixing with water, which changes the balance and causes water to flow out.
By carefully calculating the pressures exerted by each liquid column, we can precisely determine the system's response and how much water flows out of the right arm.

Fluid dynamics

Fluid dynamics is a branch of physics that deals with the behavior and movement of liquids and gases. In U-tube experiments, fluid dynamics principles help us understand why and how fluids flow from one arm to the other.
A key factor in fluid dynamics is the hydrostatic pressure, which is determined by the height and density of the liquid. This pressure dictates how fluids interact and balance within the U-tube. In our scenario, the addition of a denser liquid changes the pressure equilibrium, causing the fluid to shift and water to flow out.
By applying fluid dynamics and understanding pressure equations, we can predict how changes in liquid height and composition affect the flow in U-tube setups. This helps us calculate the exact amount of liquid displacement, giving insight into the underlying physical processes.

Density and buoyancy

Density and buoyancy are fundamental concepts that also play vital roles in U-tube experiments. Density refers to how much mass a substance contains in a given volume, while buoyancy is the upward force exerted by a fluid on any object within it.
In our experiment setup, density differences between water and the second liquid lead to pressure variations. Because the newly added liquid is less dense than water, it exerts less pressure per unit height. This density difference directly affects how liquids interact and the final equilibrium state.
The principle of buoyancy comes into play when considering how the liquids balance each other out within the tube. By calculating the densities of the involved fluids, we get a clear picture of the pressure landscape and the resultant fluid movements. This understanding allows us to solve how much water flows out when different densities and fluid heights are involved.