Solving Pendulum Equation of Motion in Cylindrical Co-ords

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In summary: Yes, you did mention the components of a vector along a given direction. However, in cylindrical coordinates, the components are expressed in terms of the unit vectors er and eθ, which are specific to the coordinate system. Therefore, the components of the weight vector along these directions are mg cos(θ) and mg sin(θ), respectively. This can be derived using the parallelogram law of forces, where the weight vector forms the diagonal of a rectangle with sides formed by the unit vectors er and eθ.
  • #1
2slowtogofast
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I am a bit hung up on this fact. Say you have a pendulum and you want to derive its equation of motion. In doing this you will have to look at the force from the weight on the end of the string. If you are using cylindrical co-ords this is what you get for Fg (sorry I don't have the diagram)

Fg = mg(cos(θ)er-sin(θ)eθ)

Where er and eθ are unit vectors r points along the direction of the string

If you were in x and y the weight has only a component in y so I am confused about when you transfer to cylindrical co-ord you get this. Can someone explain. If you do not understand what I am trying to convey I will draw a diagram when I get home today
 
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  • #2
I am not sure what is your confusion about. Changing the coordinate system will change the component of any vector, in general. Even remaining in Cartesian coordinates, the components will change if you choose the direction of the axes differently.
 
  • #3
What I want know is how to show that if I have a force in the catersian system that has only a y component. Then how is it equal to

Fg = mg(cos(θ)er-sin(θ)eθ)

In cylindrical co ords. What are the imtermediate steps? to get from one to the other?
 
  • #4
These are radial and tangential components.
For any point on the trajectory you consider the components along these two directions (radius and tangent).
 
  • #5
I attached a drawing here is what I want to know If we looked at this system in x and y W is just in y. Can someone please explain how this equation is derived.

Fg = mg(cos(θ)er-sin(θ)eθ)
 

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  • #6
Forget about the coordinate system (see footnote).

The gravitational force acts vertically. Some of that force stretches the string, some of it accelerates the weight towards the bottom position.

If you think about it, the force accelerating the weight acts tangentially to the swing: this is therefore at right angles to the force stretching the string which acts along it. To work out the components you draw a diagram like the one on this page.

You are almost there with your diagram, the vector labelled W needs to form the diagonal of a rectangle with sides formed by the vectors labelled er and etheta: this is the parallelogram law of forces.

Footnote:
Cylindrical coordinates are a 3-dimensional system which is not very helpful when considering a pendulum: the easiest 3-D system to use is the speherical system. But if we ignore the rotation of the earth, there are no forces acting outside the plane of swing so it looks like we only need a 2-D system: the 2-D equivalent of spherical coordinates is polar coordinates, and this is also the 2-D equivalent of the cylindrical system. But if we ignore streching of the string, the movement of the weight is also constrained in r, so in fact we only need to consider a single dimension, θ. There is only one 1-D coordinate system which is the linear system.
 
  • #7
2slowtogofast said:
I attached a drawing here is what I want to know If we looked at this system in x and y W is just in y. Can someone please explain how this equation is derived.

Fg = mg(cos(θ)er-sin(θ)eθ)

Find the component of W (or mg) along the direction of the string. This is mg cos(θ) and it is the component along the er.
Find the component along the direction perpendicular to the string (tangential). This is mg sin(θ) and the component along the eθ.
Do you know how to find the components of a vector along a given direction?
 
  • #8
Isn't that what I said?
 

1. How do I solve the pendulum equation of motion in cylindrical coordinates?

To solve the pendulum equation of motion in cylindrical coordinates, you will need to use the equations of motion for a pendulum, as well as the equations for cylindrical coordinates. These equations can be found in most physics textbooks or online resources.

2. What are the advantages of using cylindrical coordinates to solve the pendulum equation of motion?

Cylindrical coordinates allow for a more intuitive and straightforward approach to solving the pendulum equation of motion. It also takes into account the circular motion of the pendulum, which is not accounted for in Cartesian coordinates.

3. What are the limitations of using cylindrical coordinates to solve the pendulum equation of motion?

One limitation of using cylindrical coordinates is that it may not be suitable for all types of pendulum motion, such as when the pendulum swings in a non-circular path. Additionally, the calculations may become more complex when dealing with multiple pendulums or nonlinear systems.

4. Can the pendulum equation of motion in cylindrical coordinates be applied to real-world systems?

Yes, the pendulum equation of motion in cylindrical coordinates can be applied to real-world systems, such as pendulum clocks or swing sets. However, it is important to take into account any external forces or factors that may affect the motion of the pendulum.

5. What are some common applications of solving the pendulum equation of motion in cylindrical coordinates?

Solving the pendulum equation of motion in cylindrical coordinates can be useful in various fields, such as physics, engineering, and astronomy. It can also be used to model and analyze systems that involve circular motion, such as pendulum clocks, amusement park rides, and planetary motion.

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