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Interactive Audio Lesson
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Introduction to Basis Vectors
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Today, we will discuss basis vectors in the cylindrical coordinate system. Who can tell me what basis vectors are?
Are they the unit vectors that define a coordinate system?
Exactly! In cylindrical coordinates, we have three basis vectors: e_r, e_θ, and e_z. Let's recall their directions. Can anyone tell me what e_r represents?
e_r points radially outward from the center.
Good job! And what about e_θ?
e_θ points in the direction of increasing angle θ.
Correct! And e_z is aligned with the cylinder's axis, correct?
Yes, it's parallel to the z-axis!
Great! Remember R.E.Z. for these vectors. Radial, angular, and vertical! That's your acronym to recall their directions. Now, how do these vectors differ in behavior compared to Cartesian coordinates?
In Cartesian coordinates, the basis vectors don't change direction.
Exactly! However, in cylindrical coordinates, e_r and e_θ change as θ varies. Let's consider how changes in the radial or z direction affect our basis vectors.
I see, only changes in θ affect e_r and e_θ, not changes in r or z.
Precisely! Summarizing, we explored the definitions and behaviors of our basis vectors. Remember, R.E.Z. is key!
Visualizing Basis Vectors
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Let's visualize how e_r and e_θ change with angle θ. What happens when θ changes, Student_3?
Their directions change as θ increases.
Correct! Can you illustrate that with an example?
If we start at an angle of 0 degrees, e_r points along the x-axis, but as we increase θ to 90 degrees, e_r points up the y-axis.
Exactly! Let's draw this out. Remember the key: e_z remains unchanged while e_r and e_θ rotate around. Who can relate this back to deformation analysis?
We need to analyze forces acting on bodies that aren't fixed in direction. Understanding these changes is crucial.
Well said! So next, we'll focus on the implications of these changes on cylindrical elements used in our studies.
Implications for Analysis
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Now, let’s discuss the implications of these dynamically changing basis vectors. How do these affect our calculations?
They require us to take into account changes in direction during calculations involving stresses and strains.
Yes! Considering how e_r and e_θ change informs our analysis about forces and momentum. Can someone relate this to the cylindrical element?
In a cylindrical element, as forces act on it, we need to adjust for these changing basis vectors to maintain equilibrium.
Excellent point! When assessing elements of a cylinder under deformation, we must reference our basis vectors correctly.
So, effectively, we're bending our analysis around the concept of pivoting vectors.
Exactly, maintaining a comprehensive view leads to more accurate results. Well done! That wraps up our exploration of basis vectors!
Introduction & Overview
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Quick Overview
Standard
The section elaborates on the basis vectors of the cylindrical coordinate system, highlighting their orientation and behavior compared to Cartesian coordinates. It also explains how these vectors change with varying angles and their importance in analyzing cylindrical elements.
Detailed
Basis Vectors
This section introduces the concept of basis vectors within the cylindrical coordinate system, a critical tool for analyzing problems related to cylindrical geometries, especially in mechanics. The cylindrical coordinate system uses three basis vectors:
- e_r: Points radially outward from the center of the cylinder.
- e_θ: Points in the direction of increasing angle θ and is perpendicular to e_r.
- e_z: Aligns with the cylinder's axis of symmetry, parallel to the z-axis.
While these basis vectors are orthonormal, they differ significantly from Cartesian coordinates where the basis vectors remain constant. In cylindrical coordinates, the vectors e_r and e_θ rotate with the angle θ, changing their absolute direction while e_z remains constant.
Understanding how these basis vectors interact, especially during deformation and analyzing cylindrical elements, is fundamental in continuum mechanics. The section lays the groundwork for deriving various laws related to linear momentum balance in subsequent lectures.
Audio Book
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Introduction to Basis Vectors
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We have drawn the z=0 plane in Figure 1b. There is a baseline or reference line relative to which angle θ is measured. The length of the line joining the center and the projected point is r while the angle that it makes from the baseline is θ.
Detailed Explanation
In cylindrical coordinates, we start by observing the z=0 plane. This is the horizontal plane, where we can visualize the setup more clearly. The angle θ is measured from a reference line emanating from the center of the cylindrical coordinate system. The distance from this center to any point on the plane is denoted by r. This setup is crucial because it helps us understand how points in cylindrical coordinates are defined and measured.
Examples & Analogies
Think of a round table where you measure the distance to a guest seated at the table. The distance from the center of the table is r, and the angle from a reference line (like a pointing direction) to the guest is θ.
Components of Basis Vectors
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Two of the basis vectors lie in this plane: e_r points radially outward from the center while e_θ points in the direction of increasing θ and is perpendicular to e_r. The third basis vector e_z lies along the axis of the cylinder.
Detailed Explanation
In a cylindrical coordinate system, basis vectors are unit vectors that define directions in the system. e_r points outward from the center of the cylinder, representing the radial direction. e_θ points tangentially around the cylinder and is perpendicular to e_r, indicating the angular direction of increasing θ. Finally, e_z points vertically along the height of the cylinder, completing our set of three basis vectors, essential for describing any motion or force in this space.
Examples & Analogies
Imagine a cylindrical can. If you were to point from its center to the outer edge, that’s e_r. If you then rotated your finger around the edge of the can, that would be direction e_θ. Lastly, if you were to point straight up, that is the direction of e_z.
Comparison with Cartesian Basis Vectors
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Thus, the basis vectors for a cylindrical coordinate system are (e_r, e_θ, e_z) whereas in the Cartesian coordinate system, the basis vectors are (e_1, e_2, e_3). Although both sets of basis vectors are orthonormal, there is a big difference in their properties: the basis vectors of the Cartesian coordinate system are fixed in direction and do not change from one point to the other but in cylindrical coordinate systems, two of the basis vectors (e_r, e_θ) change when θ coordinate of a point changes.
Detailed Explanation
In Cartesian coordinates, the basis vectors e_1, e_2, and e_3 point in fixed directions along the x, y, and z axes, respectively. They remain constant regardless of the position of the point in space. In contrast, in a cylindrical coordinate system, the basis vectors e_r and e_θ change their orientations depending on the angle θ. This makes cylindrical coordinates particularly useful for problems involving circular or rotational symmetries, as it allows greater flexibility in describing motion.
Examples & Analogies
Consider how a camera positioned at a fixed point will always point in the same direction (like Cartesian coordinates). In contrast, if you were to rotate around a circular track, your direction of view would change as you moved along the track (like the basis vectors in cylindrical coordinates).
Behavior of Basis Vectors
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This has been illustrated in Figure 2a where the basis vectors at the two points are differently oriented although only the θ coordinate of the two points is different. The third basis vector e_z remains fixed at every point though. One thing to note here is that e_r and e_θ change only when the θ coordinate changes. If only the radial coordinate or/and z-coordinate of a point is changed, the basis vectors do not change.
Detailed Explanation
Figure 2a demonstrates that as you move around the cylindrical coordinate system, the orientations of e_r and e_θ vary based on the value of θ. However, e_z remains constant and does not change as you move up or down along the cylinder's height. This is an important property in understanding how forces and velocity vectors behave in cylindrical coordinates, as it indicates that radial and angular displacements affect the basis vectors, but vertical displacements do not.
Examples & Analogies
Think of a bicycle wheel. As the wheel spins, the spokes (which we can think of as similar to the basis vectors e_r and e_θ) change direction depending on where they are positioned on the wheel, but the vertical rim (akin to e_z) stays the same regardless of how the wheel rotates or moves up and down.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Cylindrical Coordinate System: Points are represented by (r, θ, z) coordinates.
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Basis Vectors: e_r, e_θ, and e_z define directional components in cylindrical systems.
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Dynamic Nature of Basis Vectors: e_r and e_θ change with angle θ while e_z remains constant.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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Example 1: If a point in cylindrical coordinates is (2, π/4, 5), e_r will point at an angle of 45 degrees from the positive x-axis.
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Example 2: In a cylindrical element experiencing radial deformation, the direction of e_θ alters based on the angle of the deformation.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
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In cylindrical ways, e_r points out the rays, e_θ twirls around, while e_z stands tall and proud.
📖 Fascinating Stories
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Imagine you're at the center of a merry-go-round. As you face outward, e_r points straight ahead. When you spin to see where you're going, e_θ rotates with you, and you never look down on e_z—that's just up!
🧠 Other Memory Gems
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Remember: R.E.Z. - Radial, in the direction of angle θ, and Vertical for the z-axis.
🎯 Super Acronyms
Use ‘RZ’ to remember Radial and z to note the fixed axis.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Basis Vectors
Definition:
Unit vectors that define direction in a coordinate system; in cylindrical coordinates these are e_r, e_θ, and e_z.
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Term: Cylindrical Coordinate System
Definition:
A three-dimensional coordinate system where points are defined by radius, angle, and height (r, θ, z).
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Term: Orthonormal
Definition:
Vectors that are both orthogonal (perpendicular) to each other and are unit vectors.
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Term: Deformation
Definition:
Change in shape or size of an object due to applied forces.