Help with coupled spring and pendulum system

[J.Sterling47]

Homework Statement

Given the system in the image below, I need to find the equation of motions for the coupled system. The surface where the block moves is frictionless. The red line is position where the block is at equilibrium. At equilbrium x1 and x2 = 0. After finding the equation of motion for the coupled system, I need to find the normal modes which I can do but I'm having trouble finding first part. Also the angle is small angle approximation.
http://imgur.com/JPzK1Jn
http://imgur.com/JPzK1Jn

Homework Equations

F= ma
sinθ = tanθ = θ = [x₂ - x₁]/L

The Attempt at a Solution

Ok so taking the horizontal and vertical forces of each:

Block:
F_x1 = -kx₁ - Tsinθ
F_y1 = N - mg - Tcosθ

Mass on Pendulum:

F_x2 = Tsinθ **************
F_y2 = mg - Tcosθ*****I feel like the spring should add a force here, but I'm not sure how.
If I proceed adding a -kx₁ to F_x2 and solving for the equations of motion using the small angle approximation I get

a₁ = -k/mx₁ + T/Lm(x₂ - x₁)

and a2 the same which is wrong. Also I cannot get rid of T. I am not so concerned over the normal modes because I can do that after. We did not do any langrange stuff so I am not allowed to use it. Also because of the small angle, I asked if I was allowed to approximate the vertical displacement for the pendulum to be zero, but I cannot so I have to find another way.

 

[OldEngr63]

You have written four equations of motion, but this system only has two degrees of freedom. You should be able to eliminate some variables after you get your equations of motion correct.

To be sure, the spring force does enter into the equation of horizontal motion for the block. How do springs generate force?

 

[J.Sterling47]

You have written four equations of motion, but this system only has two degrees of freedom. You should be able to eliminate some variables after you get your equations of motion correct.

To be sure, the spring force does enter into the equation of horizontal motion for the block. How do springs generate force?

Yeah I'm stuck here. I can't figure out how to eliminate T and I'm also unsure of how to make F_x2 correct. My problem is that I'm very limited with techniques here, as this is pretty much the beginning of an intro to classical mechanics.

I'm pretty sure adding -kx₁ to horizontal motion of the pendulum is incorrect as then the two equations that I care about are identical which is wrong. -kx₂ wouldn't make much sense.

 

[OldEngr63]

Does the spring act on the block or on the pendulum?

Where are your FBDs?

 

[HalfLostAndSearching]

I would suggest approaching this problem by considering the system as a whole, rather than breaking it down into individual forces for each component. This will allow you to use the concept of conservation of energy to find the equations of motion.

First, you can define the potential energy of the system as the sum of the potential energies of the two components:
U = 1/2kx1^2 + mgh

Next, you can define the kinetic energy of the system as the sum of the kinetic energies of the two components:
K = 1/2mv1^2 + 1/2Iω^2

Using the small angle approximation, you can express the angular velocity of the pendulum in terms of the displacement of the mass on the pendulum:
ω = (x2-x1)/L

Combining the potential and kinetic energies and taking the derivatives with respect to time, you can obtain the equations of motion for the system:
d^2x1/dt^2 = -k/mx1 + (x2-x1)/L * d^2x2/dt^2
d^2x2/dt^2 = -g/L * x2 + (x1-x2)/L * d^2x1/dt^2

These two equations, along with the initial conditions of x1 = x2 = 0 and their derivatives equal to zero at equilibrium, can be solved to find the normal modes of the system.