Question

Water is filled up to a height \(h\) in a beaker of radius \(R\) as shown in the figure. The density of water is \(\rho\), the surface tension of water is \(T\) and the atmospheric pressure is \(P_{0}\). Consider a vertical section ABCD of the water column through a diameter of the beaker. The force on water on one side of this section by water on the other side of this section has magnitude (A) \(\left|2 P_{0} R h+\pi R^{2} \rho g h-2 R T\right|\) (B) \(\left|2 P_{0} R h+R \rho g h^{2}-2 R T\right|\) (C) \(\left|P_{0} \pi R^{2}+R \rho g h^{2}-2 R T\right|\) (D) \(\left|P_{0} \pi R^{2}+R \rho g h^{2}+2 R T\right|\)

Short answer

The magnitude of the force on one side of the water column section by water on the other side is \( \left|P_{0} \pi R^{2}+R \rho g h^{2}-2 R T\right| \) (Option C).

Step-by-step solution

Identify Forces Acting on the Section

First, list all the forces acting on one side of the vertical section of water: atmospheric pressure over the surface area, the weight of the water, and the force due to surface tension.

Calculate the Force Due to Atmospheric Pressure

The force due to atmospheric pressure can be calculated as the product of the pressure and the area over which it acts, which is the surface area of the water column's top. The area is given by \( \pi R^{2} \) and the force is \( P_{0} \pi R^{2} \) because atmospheric pressure acts on the entire surface of the liquid.

Calculate the Weight of the Water Section

The weight of the water can be found by multiplying the volume of the water section by the density of water, and then by gravity (\(g\)). The volume is the surface area \( \pi R^{2} \) times the height \( h \) of the water column. The weight is \( \pi R^{2} \rho g h \) and acts at the midpoint of the height, but since we are considering forces on one side of the section, we only consider half of the height, leading to a force due to the water's weight of \( R \rho g h^{2}/2 \) when projected onto the side of section ABCD.

Calculate the Force Due to Surface Tension

Surface tension acts along the circumference of the beaker and tends to pull the water surface inwards. The total force due to surface tension is the surface tension coefficient times the length over which it acts, here the diameter of the beaker, resulting in \( 2RT \) since the diameter is \( 2R \) and there are two sides to consider.

Combine the Forces to Find the Total Force

The force by water on one side of the section by water on the other side is the sum of the forces due to pressure, the weight of the water, and surface tension. The pressure and weight exert a pushing force while surface tension exerts a pulling force. These forces are: Atmospheric pressure force \( P_{0} \pi R^{2} \), Weight of the water section \( R \rho g h^{2}/2 \) (since it's acting over a height of \( h/2 \) on the side of the section), and Surface tension force \( -2RT \) (since it's a contracting force).

Final Calculation of Force Magnitude

Adding up these forces \( (P_{0} \pi R^{2}) + (R \rho g h^{2}/2) - (2RT) \) and taking the absolute value gives us the magnitude of the force on one side of the water section. Thus, the correct answer is \( \left|P_{0} \pi R^{2}+R \rho g h^{2}/2-2 R T\right|\bounding_box{R \rho g h^{2}/2} \)

Key concepts

Atmospheric Pressure

Atmospheric pressure is the force per unit area exerted on a surface by the weight of the air above that surface in the atmosphere of Earth (or that of another planet). For any given point, it varies with the height above or below sea level, generally decreasing with increasing altitude. Most commonly, we measure atmospheric pressure in pascals or, in the context of weather reports, in millibars.

Understanding atmospheric pressure is essential in the field of fluid mechanics. It’s a significant factor in hydrostatics and fluid dynamics since it affects the behavior of fluids at rest and in motion. For instance, water’s surface in a container is subject to atmospheric pressure, influencing static fluid configurations and pressure within a fluid body.

In our exercise example, a vertical section ABCD of a water column is under the influence of atmospheric pressure, denoted by the symbol \(P_{0}\). This pressure acts on the entire surface area of the water column's top, which we calculated step-by-step as \(P_{0} \times \text{area}\). This is a fundamental concept for students to understand as it lays the groundwork for more complex calculations in fluid dynamics and related engineering applications.

Hydrostatics

Hydrostatics, also known as fluid statics, is the study of fluids at rest. The primary principle behind hydrostatics is Pascal's Law, which states that pressure exerted anywhere in a confined incompressible fluid is transmitted equally in all directions throughout the fluid. This means that any change in pressure is evenly distributed to all portions of the fluid.

Under hydrostatics, we also consider the hydrostatic pressure, which is the pressure exerted by a fluid due to its weight when at rest. It’s calculated by multiplying the fluid density (\(\rho\)), gravity (\(g\)), and the height of the fluid column (\(h\)). This concept is evident in our problem where the water's weight is a crucial factor in determining the force on a section of the water column.

For example, in our exercise improvement advice, the weight of the water was considered for only one side of the section by halving the height value to \(h/2\). This adjustment is critical to accurately calculate the force due to the weight of the water section in hydrostatic scenarios. Students should anchor their understanding of forces in fluids with the foundational principles of hydrostatics.

Fluid Dynamics

Fluid dynamics is the study of fluids (liquids and gases) in motion. It covers a wide range of applications, from flow through pipes to the aerodynamics of airplanes. The study of fluid dynamics involves understanding properties such as velocity, pressure, density, and temperature as functions of space and time.

One of the central equations in fluid dynamics is the Bernoulli's equation, which is based on the conservation of energy. It illustrates the trade-off between pressure, velocity, and height in the flow of an ideal fluid. However, our exercise involves fluids at rest, hence, the Bernoulli's equation doesn’t directly apply here, but understanding the dynamic nature of fluids helps in grasping the broader context.

Surface tension, a phenomenon closely related to fluid statics and dynamics, affects the fluid motion as it can create additional pressure on the fluid’s surface. In our problem, the force due to surface tension acts along the circumference of the beaker's top edge and plays a role in the total force balance. It is essential for students to recognize the conditions under which the principles of fluid statics overlap into fluid dynamics to better understand fluid behavior.