Cartesian to Cylindrical coordinates?

  • #1

Homework Statement


I want to convert R = xi + yj + zk in to cylindrical coordinates and get the acceleration in cylindrical coordinates.

Homework Equations


Inline31.gif
Inline32.gif
Inline33.gif

Inline34.gif
Inline35.gif
Inline36.gif

Inline37.gif
Inline38.gif
z

The Attempt at a Solution


I input the equations listed into R giving me:

R =
Inline33.gif
i +
Inline36.gif
j + z k

Apply chain rule twice:

Inline209.gif


The final answer is:

Inline212.gif


How do I get this final answer? It looks like the terms with sin were dropped. How does this happen?
 

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Answers and Replies

  • #2
Orodruin
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Your result is an expression for the acceleration using the Cartesian vector basis (i.e., you are showing the Cartesian components expressed in terms of the cylinder coordinates). You need to relate this to the vector components using the cylinder coordinate basis vectors.
 
  • #3
Your result is an expression for the acceleration using the Cartesian vector basis (i.e., you are showing the Cartesian components expressed in terms of the cylinder coordinates). You need to relate this to the vector components using the cylinder coordinate basis vectors.
I don't think I understand. By basis you mean the unit vectors in rHat, thetaHat, zHat?
 
  • #4
Orodruin
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Yes, that is the basis that you should be using to express your vector as done in the quoted result.
 
  • #5
Yes, that is the basis that you should be using to express your vector as done in the quoted result.
So I have:

Inline69.gif
Inline70.gif
Inline71.gif

Inline72.gif
Inline73.gif
Inline74.gif

Inline75.gif
Inline76.gif
Inline77.gif


I don't see how I replace i,j,k with these to get the answer.
 

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  • #6
Orodruin
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You have a result for your vector. You need to express it as a linear combination of the vector basis, i.e., you need to find ##v_r##, ##v_\theta## and ##v_z## such that
$$
\vec v = v_r \hat r + v_\theta \hat \theta + v_z \hat z.
$$
Since you have three components, this is a system of three equations for three unknowns.
 
  • #7
You have a result for your vector. You need to express it as a linear combination of the vector basis, i.e., you need to find ##v_r##, ##v_\theta## and ##v_z## such that
$$
\vec v = v_r \hat r + v_\theta \hat \theta + v_z \hat z.
$$
Since you have three components, this is a system of three equations for three unknowns.

I see it. Thanks!
 

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