Podcast
Questions and Answers
To convert rectangular coordinates to polar coordinates, the formula for r is r = √(x² + ______²)
To convert rectangular coordinates to polar coordinates, the formula for r is r = √(x² + ______²)
y
In polar coordinates, the angle θ is given by θ = tan⁻¹(y/______)
In polar coordinates, the angle θ is given by θ = tan⁻¹(y/______)
x
The rectangular coordinates can be found from polar coordinates using x = r cos ______
The rectangular coordinates can be found from polar coordinates using x = r cos ______
θ
For forces, the resultant force F is found using F = √(Fx² + ______²)
For forces, the resultant force F is found using F = √(Fx² + ______²)
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In polar form, a force vector is represented as F = (F, ______)
In polar form, a force vector is represented as F = (F, ______)
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A coordinate system gives us a frame of reference to describe a system that we would like to analyze in ______.
A coordinate system gives us a frame of reference to describe a system that we would like to analyze in ______.
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In an orthogonal coordinate system, the coordinate directions are ______ to each other.
In an orthogonal coordinate system, the coordinate directions are ______ to each other.
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In the Cartesian system, the two coordinate axes are named ______ and y.
In the Cartesian system, the two coordinate axes are named ______ and y.
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Points are specified as an ordered pair of coordinate values enclosed in ______.
Points are specified as an ordered pair of coordinate values enclosed in ______.
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In the polar coordinate system, a point is specified by its distance from the origin, ______, and an angle measured counter-clockwise.
In the polar coordinate system, a point is specified by its distance from the origin, ______, and an angle measured counter-clockwise.
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In polar coordinates, points are specified as an ordered pair of values separated by a ______.
In polar coordinates, points are specified as an ordered pair of values separated by a ______.
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To transform between rectangular and polar coordinate systems, the formulas convert (x, y) into ______, θ.
To transform between rectangular and polar coordinate systems, the formulas convert (x, y) into ______, θ.
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The intersection of the coordinate axes is called the ______.
The intersection of the coordinate axes is called the ______.
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Study Notes
2D Coordinate Systems & Vectors
- A coordinate system provides a reference frame for describing and analyzing systems.
- Orthogonal coordinate systems are commonly used in statics. Orthogonal means perpendicular.
- Coordinate directions in an orthogonal system are perpendicular to each other, making them independent.
- The intersection of coordinate axes is the origin.
- Measurements are made from the origin.
- Points and vectors are described using a set of numbers called coordinates.
- Two-dimensional orthogonal coordinate systems include rectangular and polar coordinates.
Rectangular Coordinates
- The Cartesian system is the most important rectangular coordinate system.
- It has two coordinate axes (x and y) at 90-degree angles.
- Typically, the x-axis is horizontal and points to the right, the y-axis is vertical and points upward.
- Points are represented as ordered pairs (x, y).
- Forces and other vectors are represented using scalar components (e.g., F = (Fx, Fy)).
Polar Coordinates
- An alternative orthogonal system.
- A point is described by its distance (r) from the origin and an angle (θ) measured counter-clockwise from a reference direction (usually the positive x-axis).
- Points are represented as ordered pairs (r, θ).
Coordinate Transformations
- Rectangular coordinates can be converted to polar coordinates, and vice versa using specific formulas.
- Formulas are provided for converting rectangular coordinates to polar (r=√(x²+y²); θ= tan⁻¹(y/x)).
- Formulas are provided for converting polar coordinates to rectangular (x=r cos(θ); y= r sin(θ)).
- These transformations are useful for analyzing forces and points in different coordinate systems.
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Description
This quiz covers the fundamentals of two-dimensional coordinate systems, including rectangular and polar coordinates. It explores their significance in statics and how to represent points and vectors using these systems. Understanding orthogonal coordinate systems is key to mastering these concepts.
