- #1

Sefrez

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## Homework Statement

The problem states that a pendulum is attached to a spring that can only oscillate in the vertical direction. I am supposed to derive equations of motion for the spring/pendulum system under my chosen generalized coordinates.

## Homework Equations

Lagrange equations of motion

## The Attempt at a Solution

I choose a to describe the system by the height of the pendulum mass and the angle at which it swings. So the two coordinates are y and θ, respectively.

Given a spring constant, k, and the assumption of a massless spring, I formulated the potential energy of the system to be:

U = (1/2)ky^2+mg(y-l*cos(θ))

where l is the length of the pendulum, and I have chosen the potential to be zero at y = 0 & θ = pi/2.

The kinetic energy of the system should be that given only by the mass of the pendulum. I formulated this to be:

T = (1/2)m((vx)^2 + (vy)^2) = (1/2)m((l*θ'*cos(θ))^2 + (y' + l*θ'*sin(θ))^2)

Without attempting to put up all the math on the board, I computed the Lagrange equations on y and θ, and found the two differential equations:

y'' + (k/m)y +l*θ''*sin(θ) + l*(θ')^2*cos(θ)+g = 0

and

θ'' + (1/l)y''*sin(θ) + (1/l)*g*sin(θ) = 0

I am unsure if I have made any mistakes, but am I going about this problem at all correctly?

Thanks.