# Pendulum consists of a rod of mass m attached to a light rod

1. Aug 28, 2012

### cler

1. The problem statement, all variables and given/known data
A pendulum consists of a uniform rod of mass m and length l hanging from the bottom end of a light rod of length l which top end is fixed to the ceiling. (see file attached)
System moves in a vertical plane. Find equations of motion.

Coordinates of the center of mass (X,Y)
angles θ and ψ of the light rod and the rod of mass m with the vertical respectively.

2. Relevant equations

Lagrangian method

L=T-U
U=mgY
T=mV2/2 + IΩ2/2

V is the velocity of the center of mass respect to a system at rest which origin is the top end of the light rod.

3. The attempt at a solution

X=l/2sinψ + lsinθ
Y=-l/2cosψ -lcosθ

|V|2= l2/4$\dot{ψ}$ + l2$\dot{θ}$2 + l2$\dot{ψ}$$\dot{θ}$cos(ψ-θ)

I relative to the top end of the rod of mass m I=ml2/3
ω=$\dot{ψ}$

then i will plug this into L= T-U and the fin the Euler- Lagrange equations.
but i am not sure about I and ω.

I am confused. My first attempt was to choose same X,Y,V but I relative to the center of the rod of mass m I= ml2/2 and ω=$\dot{ψ}$+$\dot{θ}$

#### Attached Files:

• ###### DSC_0318.jpg
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Last edited: Aug 28, 2012