Calculating Pencil Pendulum Angle with X/Y Coords

[irrationalistic]

I have a question that hours of internet research have not answered. I noticed that, for example, when I am holding a pencil by the eraser end and I let gravity hold it in a vertical position, when i move my hand back and forth, the tip of the pencil lags behind then forms a pendulum with that new angle. What i am looking for is, on a cartesian plane, if given the x and y coordinates of the eraser end of the pencil and the velocity at which that end is moving, if there is an equation to calculate the new angle formed.
Thanks!

 

[Mr.Socrates]

see when the pencil moves in a pendulum it forms a complex pendulum in yz axis and since the pencil itself moves in xy axis it follows a equation with x and y coordinates in sinusoidal function... the pencil if u consider as a whole has many points and has a single Centre of mass the motion of those points form a matrix of equations with yz coordinates and if u have time and leisure just solve those equations and get a unified solution in cartesian system which each point of ur pendulum will follow

 

[arildno]

This is one possible way of doing this:
1.Suppose you move your hand back and forth with acceleration a(t) in the positive x-direction, a known function.

2. Go into the rest frame of the eraser end.
In this frame, there's three "forces" acting upon the pencil:
a) The weight of the pencil, acting at the center of mass of the pencil
b) The force from your hand, acting on the eraser end
c) The auxiliary force, -ma(t), acting at the center of mass of the pencil

3. Take the moment with respect to the eraser head in order to eliminate the contact force b) from further consideration.

The resulting moment of momentum equation gives you the differential equation for the angular displacement.

 

[fleon]

I think, the last answer was formulated correctly, but however, you must put the acceleration a(t) in the positive x-direction like a function X=ACos(wt+Fi). In that way the solution of your problem is very simple.

 

[Mk]

I have a question that hours of internet research have not answered. I noticed that, for example, when I am holding a pencil by the eraser end and I let gravity hold it in a vertical position, when i move my hand back and forth, the tip of the pencil lags behind then forms a pendulum with that new angle.

What do you mean the tip of your pencil lags behind?