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I am working through some problems from the book Classical Dynamics of Particles and Systems by Marion and Thornton, more specifically Problem 7-18. The question is as in the picture.enter image description here

My attempt at the solution is the following. I found the Lagrangian to be

$L=\frac{1}{2}m\dot{\phi}^2(l-R\phi)^2-mg(R\cos(\phi)-(l-R\phi)\sin(\phi))$,

where I have used the coordinate $\phi$ to be the angle from the uppermost point of the disk to the point the string stops touching the disk. I used this in the Euler-Lagrange equation and found the equation of motion to be

$\ddot{\phi}=\frac{1}{l-R\phi}(R\dot{\phi}^2+g\cos(\phi))$.

Then, to find the frequency of small oscillation, I assumed that $\phi=\phi_0+\eta$, where $\eta$ is a small variation of the angle and $\phi_0$ is the equilibrium angle. I plugged this in the equation of motion and (after some rearranging and a couple of first-order Taylor expansions) arrived to

$0=\ddot{\eta}(l-R\phi_0)-R(\ddot{\eta}\eta+\dot{\eta}^2)-g(\cos(\phi_0)-\sin(\phi_0)\eta)$,

or

$0=\ddot{\eta}(l-R\phi_0)-R\frac{d}{dt}(\dot{\eta}\eta)-g(\cos(\phi_0)-\sin(\phi_0)\eta)$.

Here comes my specific question. According to the solution given on the back of the textbook, it seems that the term with $d(\dot{\eta}\eta)/dt$ has to somehow vanish, but I don't see how is this possible. I assume that it probably has to come from the fact that $\eta$ is small, but even then I do not see how such conclusion can be justified.

As an extra remark, the solution states that

$\omega =\sqrt{\frac{g\sin(\phi_0)}{l-R\phi_0}}$

is the frequency of small oscillation.

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If the oscillation is small, then $\eta(t)=A\cdot f(t)$, where $A$ is the (small) amplitude of oscillation and $f(t)$ is a sinusoidal function of some finite frequency $\omega$. The products $\eta\eta$, $\dot{\eta}\eta$, $\ddot{\eta}\ddot{\eta}$, $\ldots$ are all of the form $A^2\cdot\omega^n\cdot g(t)$, where $\omega^n$ is finite and $g(t)$ is sinusoidal. Hence they are still second order in amplitude and can be neglected in an analysis of small oscillations.

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