# Pendulum Attached to Uppermost Point of Vertical Disk

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.

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.

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.