Problem 83 - Fundamentals of Physics

Question

Shows a siphon, which is a device for removing liquid from a container. Tube \(A B C\) must initially be filled, but once this has been done, liquid will flow through the tube until the liquid surface in the container is level with the tube opening at \(A\). The liquid has density \(1000 \mathrm{~kg} / \mathrm{m}^{3}\) and negligible viscosity. The distances shown are \(h_{1}=25 \mathrm{~cm}, d=\) \(12 \mathrm{~cm}\), and \(h_{2}=40 \mathrm{~cm} .\) (a) With what speed does the liquid emerge from the tube at \(C ?(b)\) If the atmospheric pressure is \(1.0 \times 10^{5} \mathrm{~Pa}\), what is the pressure in the liquid at the topmost point \(B ?\) (c) Theoretically, what is the greatest possible height \(h_{1}\) that a siphon can lift water?

Step-by-Step Solution

Answer

a) 2.21 m/s; b) 96076 Pa; c) 10.2 m

Step 1: Understanding the problem

We are dealing with a siphon, which will expel liquid once initially filled. We will calculate the exit speed, the pressure at a certain point, and the maximum possible lift height of water using given distances and densities.

Step 1: Determine the velocity at point C

Using Bernoulli's equation, the velocity of liquid emerging at the end of the siphon can be found as:\[v_c = \sqrt{2gh_1}\]Where \(g = 9.81 \, m/s^2\) is the acceleration due to gravity, and \(h_1 = 0.25 \, m\); the speed at \(C\) is:\[v_c = \sqrt{2 \times 9.81 \, m/s^2 \times 0.25 \, m} = \sqrt{4.905} = 2.21 \, m/s\].

Step 2: Calculate the pressure at the topmost point B

At point B, the height above the reservoir surface is \(h_2 = 0.40 \, m\). The pressure at B can be calculated using Bernoulli’s principle:\[P_b = P_a - \rho g h_2\]where \(P_a = 1.0 \times 10^5 \, Pa\) (atmospheric pressure), \(\rho = 1000 \, kg/m^3\), and \(g = 9.81 \, m/s^2\). Substituting,\[P_b = (1.0 \times 10^5) - (1000 \times 9.81 \times 0.40) = 1.0 \times 10^5 - 3924\]\[P_b = 96076 \, Pa\].

Step 3: Determine the greatest possible height for a siphon

Theoretically, the greatest height \(h_1\) a siphon can lift water is limited by atmospheric pressure. If the siphon lifts water higher than this, a vacuum will form, breaking the siphon.According to Bernoulli's equation, if we raise the fluid to maximum height, velocity becomes 0:\[P_a = \rho g h_{max}\]where \(P_a = 1.0 \times 10^5 \, Pa\) and rewriting,\[h_{max} = \frac{1.0 \times 10^5}{1000 \times 9.81} \approx 10.2 \, m\].

Key Concepts

Bernoulli's equation

Bernoulli's equation is a fundamental principle in fluid dynamics, connecting various properties of moving fluids. It allows us to calculate the speed of fluid flow and understand pressure changes within a fluid. The equation is written as: \[ P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant} \]where:

• \(P\) is the fluid pressure.
• \(\rho\) is the fluid density.
• \(v\) is the flow velocity.
• \(h\) is the height in relation to some reference point.
• \(g\) is the acceleration due to gravity.

This formula emphasizes how change in pressure results from changes in speed and height of the fluid. In the context of a siphon, by substituting known values into the equation, we can resolve for unknowns like flow velocity, as demonstrated in the problem at hand.

fluid dynamics

Fluid dynamics is the study of fluid behavior in motion, encompassing various laws and principles—in particular, Bernoulli's equation. In a siphon, fluid dynamics helps explain how liquid flows unaided from a container using gravity and pressure differences. The key components in fluid dynamics include:

• The continuity equation, which ensures fluid mass is conserved throughout its flow path.
• Archimedes' principle, which explains how buoyancy affects fluid displacement.
• Bernoulli's principle, essential in calculating kinetic energy changes in the fluid.

Visualizing a siphon helps in understanding these principles: Initially, the tube needs filling to create an uninterrupted flow. Once the flow is established, atmospheric pressure and gravity work together to sustain it, allowing the fluid to siphon independently.

atmospheric pressure

Atmospheric pressure is the force exerted by the weight of the air column above a surface, measured in Pascals (Pa). At sea level, standard atmospheric pressure is about \(1.0 \times 10^5 \, \text{Pa}\). In the context of a siphon, atmospheric pressure plays a crucial role.It's this pressure that pushes the liquid up and over the bend in the siphon, allowing it to flow downwards. Without sufficient atmospheric pressure, a siphon wouldn't work as the liquid would fail to rise to the needed height. Knowing atmospheric pressure helps in calculating the maximum height a siphon can lift water. When the siphon tube's height surpasses a certain limit, a vacuum can form in the highest section, disrupting the flow and stopping the siphon.

velocity calculation

Calculating the velocity of a fluid at different points in a siphon can be accomplished using Bernoulli's equation. The equation simplifies to calculate velocity as:\[ v = \sqrt{2gh} \]where \(h\) is the height difference between two points in a fluid.For a siphon, since friction and other forces are negligible, the formula provides an accurate estimation of speed. Understanding this calculation allows for determining not just exit velocity but also informs decisions about siphon design and functionality, ensuring efficient fluid transfer.