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Interactive Audio Lesson
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Introduction to Force on +r and −r Planes
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Today, we will start discussing the forces acting on the +r and −r planes of a cylindrical element. Can anyone recall what we discussed about cylindrical coordinates in previous lectures?
We learned that cylindrical coordinates are useful for dealing with problems involving circular symmetry.
Exactly! Now, we will apply this understanding to compute the forces. Recall that the assumption we will use is that the traction is constant across the plane. Does anyone know why this assumption is helpful?
It simplifies the calculations because we do not have to analyze variations in traction.
Correct! Using the constant traction assumption allows us to accurately compute total forces efficiently. We can express the total force on the +r plane as the traction multiplied by the plane area. This is a critical step in our analysis.
Calculating Force on +r and −r Planes
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Let's calculate the forces now. The area of the +r plane can be found as the product of its height and curved edge. Who can remind us of the formula for the area A?
Is it A = r∆θ∆z for the z planes?
Close! For the +r plane, we need to consider the curvature. The area is actually calculated as the curved edge multiplied by the height. Here, it will be related to ∆r instead.
So we multiply the traction at the center with the area to find the total force?
Exactly! And once we calculate the forces on both +r and −r planes, we can sum them up to understand the net effect and how it influences momentum balance.
Understanding Taylor's Expansion
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We also briefly mentioned Taylor's expansion earlier. Can someone explain its role in our derivations?
It expands functions to account for changes in variables, right?
That's correct! In this context, it helps us approximate the variation in traction as we observe the cylindrical element in different planes.
But we don't actually need to perform the full calculation because our assumption simplifies the process?
Exactly! The Taylor's expansion confirms that smaller order terms drop out when averaged over the volume. Thus, we can effectively use our simplified models without compromising accuracy.
Final Calculation of Forces
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Now let's bring everything together. Using the formulas we discussed, how would you combine the forces from the +r and −r planes?
We add them since they both act on the same cylindrical element!
Correct! The final expression gives us an insight into how forces act within the cylindrical coordinate system. Understanding these forces helps lay the groundwork for the Linear Momentum Balance moving forward.
So this will influence our next topics about momentum balances too?
Absolutely, and this is why mastering these concepts is crucial for tackling subsequent subjects.
Introduction & Overview
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Quick Overview
Standard
In this section, the total forces acting on the +r and −r planes are derived using an approximate method. The concept of constant traction at the center of the planes is introduced, resulting in a simplified expression for calculating forces in cylindrical coordinates.
Detailed
Detailed Summary
This section delves into calculating the total force exerted on the +r and -r planes of a cylindrical element by using an approximate method rather than more complex derivations. The importance of understanding these forces is tied to their roles in the Linear Momentum Balance (LMB).
Key Points Covered:
- Assumption of Constant Traction: The derivation assumes that the traction on the +r and −r planes is constant, equating it to the value at the center of the respective planes. This simplifies calculations.
- Area Calculation: The area of each plane is determined by multiplying the curved edge of the cylinder by its height, leading to a straightforward formula for total forces.
- Force Calculation: The total forces acting on both planes are derived by multiplying the traction at the center of the planes by their respective areas. This straightforward approach reduces complexity and streamlines the calculations in the context of cylindrical coordinates.
- Taylor's Expansion: The Taylor series expansion is briefly discussed to illustrate the assumptions made about variations in stress components across the planes. However, it ultimately shows that assuming constant traction simplifies the mathematics without sacrificing accuracy.
This foundational understanding is crucial as it sets the stage for future discussions on momentum balance in more complex cylindrical phenomena.
Audio Book
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Assumptions for Traction on +r and −r Planes
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Forgetting the total force on +r and −r planes, we will use the approximate method just discussed above as the exact derivation is much more tedious. We thus assume that traction on the +r plane (the convex plane) is constant and equal to the value at its center, i.e. at (r + ∆r, θ, z) and also the traction on the −r plane is constant and equal to the value at (r - ∆r, θ, z).
Detailed Explanation
In this step, we introduce the basic premise for calculating the forces on the +r and -r planes of a cylindrical element. Instead of accounting for variations of traction across these planes, we simplify the calculation by assuming that the traction value is uniform across the surface. This allows us to treat it as constant, which simplifies the calculations and helps establish a clear starting point for deriving total forces.
Examples & Analogies
Imagine you have a balloon. When you blow air into it, the pressure inside is relatively consistent if you don't blow too hard. In this scenario, assuming a constant pressure throughout the surface of the balloon simplifies how we calculate the total force being exerted on the balloon's surface.
Calculating the Area of the +r and −r Planes
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We will denote the area of the +r plane by A and the area of the −r plane by A-. From Figure 2, we can observe that the area of the r planes will be equal to the curved edge multiplied with the height.
Detailed Explanation
Here, we define the areas of the two planes, +r and -r. The area of each plane can be determined by multiplying the circumference of the curved surface (which is dependent on radius) by the height of the cylindrical section. This geometric relationship is crucial for later calculations of force, as the total force will depend on both the traction at the center of each plane and the area over which it acts.
Examples & Analogies
Think of wrapping a label around a soft drink can. The area of the label (analogous to the area of the curved surface here) depends on the height of the can and the distance around it, which is related to the radius. Knowing how much space you have for labeling helps you understand the total area available for any stickers or branding.
Expression for Total Force on +r and −r Planes
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The forces on both the planes can be simply obtained by multiplying the traction at the center of the planes with the area of the planes. So, the total force on +r and −r planes will be...
Detailed Explanation
In this section, we derive the total force exerted on the +r and -r planes by multiplying the traction (the force per unit area) at the center of each plane by the area of these planes. This stems from the fundamental physics principle that total force equals pressure multiplied by area. Thus, using our previous assumptions, we can obtain an elegant expression for the forces acting upon these planes.
Examples & Analogies
Imagine you're pushing against a wall with your hand. The force you exert can be thought of as pressure distributed over the area of your hand that is in contact with the wall. If you press harder (increasing pressure) or if your hand covers a larger area (increasing area), the total force you exert will be greater. This example illustrates how area and pressure combine to define force.
Basis Vectors and Taylor Expansion
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The traction vectors can be written as... The basis vectors at the centers of the r planes are the same as those at the center of the cylindrical element as they have the same θ coordinate. Thus, we would not have to use Taylor’s expansion for basis vectors.
Detailed Explanation
In this chunk, we acknowledge that the traction vectors' orientation is consistent with the direction of the cylindrical element's core. Since the basis vectors remain unchanged when moving between the center of the element and the +r or -r planes, we avoid the complexity of using Taylor's expansion to evaluate variations of these vectors. This simplification is a key part of our calculation, allowing us to focus on the effect of traction changes more directly.
Examples & Analogies
Think of how you often use the same direction to point to different objects in a room. If your direction of focus doesn't change—like your perspective towards an array of shelves—you can easily describe the distance without having to adjust your orientation or perspective. Likewise, in this scenario, the vectors don’t change, making it simpler to compute.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Constant Traction: The assumption that traction on the +r and −r planes is constant significantly simplifies force calculations.
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Area Calculation: The area of the planes is determined based on the height and curvature, which is key for determining total forces.
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Taylor's Expansion: Using Taylor's expansion helps understand how variations in traction can affect modeling but is often unnecessary for simple calculations.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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An example of the force calculation on the +r plane can be derived by assuming a traction of 10 N/m² at the center, with an area measurement of 5 m², leading to a force of 50 N.
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When considering different cylindrical elements, such as those found in piping systems, similar assumptions regarding traction can be applied for pressure forces.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
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In cylindrical shape, forces give way, +r and -r pull, through each day.
📖 Fascinating Stories
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Imagine a giant soda can. The pressure pushes outward equally on the curved parts, maintaining balance on the +r and -r.
🧠 Other Memory Gems
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P.A. (Pressure times Area) will lead to force — think of a flat tire needing push to go on course.
🎯 Super Acronyms
CAP (Constant Area Pressure) gives us the way to figure forces in cylindrical play.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Cylindrical Coordinates
Definition:
Coordinate system that uses three values (radial distance, angle, height) to define points in three-dimensional space.
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Term: Traction
Definition:
Force per unit area exerted on a surface, critical for understanding stress analysis.
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Term: Taylor Series Expansion
Definition:
Mathematical representation that approximates functions near a specified point.
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Term: Area of a Plane
Definition:
Computed as the product of dimensions relevant to the shape being analyzed; in this context, the curved edge times height.