Vibration of pulley and string system So here's the statement:
A pulley of a mass $M$ is hanged using a spring (stiffness of the string being $k_1$), as shown in the image. What is the frequency of the pulley's oscillation?

So that's as far as I could get:
Linear motion equation:
$$T_2+T_1-mg=ma \tag{1}$$
Where $T_2$ is string tension on the spring side ($T_2=-kx$) and $T_1$ being string tension on the other side.
Now for the rotational equation:
$$R(T_2-T_1)=I\alpha \tag{2}$$
$$a=R\alpha\tag{3}$$
$$I=\frac{mR^2}{2}\tag{4}$$
Now I have a feeling that the problem should be solved using the equations:
$$x'' + w^2x=0$$
$$T=\frac{2\pi}{w}$$
In this case $x''=a$. From the equations (2), (3) and (4) we derive that
$$T_2-T_1=\frac{ma}{2}\tag{5}$$
And adding (5) and (1) we get
$$-2T_2+mg+\frac{3}{2}ma=0$$
or
$$2kx+mg+\frac{3}{2}ma=0$$
And there I'm stuck. Could anyone tell if at least I'm going to the right direction? Any help is appreciated!
 A: To get to natural frequency with 1DOF you need to work out the equations in the form
$$ \ddot{x} = -\omega^2 x $$
where $x$ is the displacement from the static equilibrium and $\omega$ the frequency in $\rm rad/s$. To get to frequency in $\rm Hz$ you take $f=\frac{\omega}{2\pi}$.
A: 
Please refer the diagram shown for reference points.
Dude, before jumping into all these equations try to have a feel of what's gonna happen. 
Since the string on the right hand side is inextensible, the maximum possible motion in the vertical direction is piR where R is the radius of the pulley. Also, if  we assume frictional force exists between the spring/pulley or rope/pulley, the pulley will also rotate & otherwise it will not. In case of perfectly smooth surface, the pulley will execute Simple Harmonic motion i.e. point B will move to point C [assuming sufficient M], C will move upwards by piR. Then it will again revert to its initial state.   
However, if friction, a non-conservative force, exists, along with downward motion, the pulley will also rotate about its centre. In addition, it will also execute simple harmonic motion, but a damping term will also figure in your governing equation i.e. after a certain time [which will be given by the time-constant of the equation], the oscillations become barely noticeable.
Going back to equations,
the first one you wrote,
T2+T1-mg=ma
is ok.
T2=-kx
a=d2x/dt2
So it gives a second-order ODE in x w.r.t. t.
But the second one,
for frictionless pulley, alpha=0
So it cannot be written.
Similarly the third one,
is the tangential/rotational acceleration & not the vertical a which you have used in the first.
Hope this helps.
