Solving Pendulum Equation of Motion in Cylindrical Co-ords

[2slowtogofast]

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)e_r-sin(θ)e_θ)

Where e_r and e_θ 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

 

[nasu]

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.

 

[2slowtogofast]

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?

 

[nasu]

These are radial and tangential components.
For any point on the trajectory you consider the components along these two directions (radius and tangent).

 

[2slowtogofast]

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)

 

[pbuk]

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.

 

[nasu]

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?

 

[pbuk]

Isn't that what I said?