- #1
tiago23
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Velocity in polar coordinates (again)
- Thread starter tiago23
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SUMMARY
The discussion centers on understanding the derivation of velocity in polar coordinates, specifically regarding Equation 1.11.3. Users are encouraged to resolve the unit vectors ##e_r## and ##e_{\theta}## into Cartesian components, expressed as $$e_r=e_x\cos{\theta}+e_y\sin{\theta}$$ and $$e_{\theta}=-e_x\sin{\theta}+e_y\cos{\theta}$$. The confusion arises from the relationship between the magnitudes of Δer and Δθ, which requires a clear understanding of vector differentiation in polar coordinates. The conversation highlights the need for clarity in the derivation process to grasp the underlying concepts.
PREREQUISITES- Understanding of polar coordinates and their applications in physics.
- Familiarity with vector decomposition in Cartesian coordinates.
- Knowledge of differentiation with respect to time in the context of vector calculus.
- Basic grasp of trigonometric functions and their role in vector representation.
- Study the derivation of velocity in polar coordinates using vector calculus.
- Learn about the implications of unit vector transformations between polar and Cartesian coordinates.
- Explore advanced topics in vector differentiation, focusing on time-dependent changes.
- Review related physics concepts, such as angular velocity and its relation to linear velocity in polar systems.
Students and educators in physics, particularly those focusing on mechanics and vector calculus, as well as anyone seeking to deepen their understanding of polar coordinate systems and their applications in motion analysis.
- #2
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The easiest way to show this is to first resolve ##e_r## and ##e_{\theta}## into components in the cartesian coordinate directions:
$$e_r=e_x\cos{\theta}+e_y\sin{\theta}$$
$$e_{\theta}=-e_x\sin{\theta}+e_y\cos{\theta}$$
Are you OK with this so far?
$$e_r=e_x\cos{\theta}+e_y\sin{\theta}$$
$$e_{\theta}=-e_x\sin{\theta}+e_y\cos{\theta}$$
Are you OK with this so far?
- #3
tiago23
- 4
- 1
Hey @Chestermiller sorry it took so long for me to reply, internet access here is a bit precarious. I understand the derivation of this relation first decomposing the vectors into its components, and then differentiating it with relation to time, but the approach taken in the book (and the answer to the previous question) is the one I can't wrap my mind around. Sorry if this a bit capricious or demanding, I just want to understand the approach taken. :)
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