Using polar coordinates in 1-dimensional problems

In summary, choosing a coordinate system can greatly affect the convenience of solving a physical problem. In the case of a particle constrained to move in the y direction, writing y in terms of polar coordinates may seem like a good idea, but it actually makes the problem harder to solve as only one of the polar coordinates will be a free parameter. This can be seen in the example of the line x=1, where the relation between y and θ is just a non-linear change of parametrisation.
  • #1
kent davidge
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If I have a physical problem, say, a particle which is constrained to move in the ##y## direction, which means that its ##x## coordinate remains fixed, does it make sense to write ##y## in terms of polar coordinates? That is, ##y = r \sin\theta##. Since now I have two parameters ##r,\theta## varying, so this has only made solving the problem harder than it must be, right?
 
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  • #2
kent davidge said:
so this has only made solving the problem harder than it must be, right?
Yes.
 
  • #3
We choose coordinate systems to be convenient. Some choices can be very inconvenient.
 
  • #4
kent davidge said:
If I have a physical problem, say, a particle which is constrained to move in the ##y## direction, which means that its ##x## coordinate remains fixed, does it make sense to write ##y## in terms of polar coordinates? That is, ##y = r \sin\theta##. Since now I have two parameters ##r,\theta## varying, so this has only made solving the problem harder than it must be, right?
If you do this, only one of your polar coordinates will be a free parameter as the other will be fixed. For example, consider the line ##x = 1##. On this line, ##r^2 = 1 + y^2## which means that, if you fix ##\theta##, then ##r^2(1-\sin^2\theta) = 1## and so ##r = 1/\cos\theta##. The relation between ##y## and ##\theta## is therefore ##y = \tan\theta##, which is just a (non-linear) change of parametrisation of your line.
 
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Related to Using polar coordinates in 1-dimensional problems

What are polar coordinates?

Polar coordinates are a system of representing points in a plane using a distance from the origin and an angle from a fixed reference direction.

How are polar coordinates used in 1-dimensional problems?

In 1-dimensional problems, polar coordinates are used to represent points along a single axis, typically the x-axis. This allows for a more efficient and intuitive way of solving problems involving circular or rotational motion.

What are the advantages of using polar coordinates in 1-dimensional problems?

One advantage of using polar coordinates in 1-dimensional problems is that it simplifies the equations and makes them easier to solve. It also allows for a more visual representation of the problem, making it easier to understand and analyze.

Are there any limitations to using polar coordinates in 1-dimensional problems?

Yes, there are some limitations to using polar coordinates in 1-dimensional problems. It is not suitable for all types of problems and may not be as accurate as other coordinate systems in certain situations.

How can I convert between polar coordinates and Cartesian coordinates?

To convert from polar coordinates to Cartesian coordinates, you can use the formulas x = r * cos(theta) and y = r * sin(theta). To convert from Cartesian coordinates to polar coordinates, you can use the formulas r = sqrt(x^2 + y^2) and theta = arctan(y/x).

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