Solve Ehrenfest's Pendulum Equation | Can't Solve It

[dRic2]

Homework Statement

A simple pendulum hangs from a fixed pulley. The other end of the string is in the hand of an observer who pulls up the string slowly, thus shortening the length of the pendulum with uniform velocity. Show that, neglecting friction, the amplitude of the oscillations increases in the following manner. The change of the total energy from the position $\theta = 0$ to the next position $\theta = 0$ is given by $$\Delta E = - \frac 1 2 \frac {\Delta l} l E$$ where $E$ is the energy constant of the undisturbed oscillations.

Relevant Equations

$$\Delta E = - \int_{t_1}^{t_2} \frac {\partial L} {\partial t} dt$$

Well, using the above equation it should be easy... but I can't solve it

$$ L = \frac 1 2 m (\dot l^2 + l^2 \dot \theta ^ 2) - mgl(1- \cos\theta)$$

then I guess

$$\int_{t_1}^{t_2} \frac {\partial L}{\partial t} dt = L(t_2) - L(t_1)$$

*Note*: since the variation $\frac {\partial L}{\partial t}$ is considered wrt to the variable $t$ alone, in the infinitesimal time $dt$, $\theta$ and $\dot \theta$ are kept constant. So, even when integrating $\theta$ and $\dot \theta$ are kept constant wrt to time. Am I right ? If I choose $t_2$ to be the instant when $\theta$ is again equal to zero and, of course, $\theta(t_1) = 0$ then:

$$\Delta E = -\frac 1 2 m (\dot l^2 + l^2(t_2) \dot \theta ^2) - \frac 1 2 m (\dot l^2 + l^2(t_1) \dot \theta ^2)$$

and $l^2(t_2) = (l(t_1) + \Delta l)^2 \approx l^2(t_1) + 2l \Delta l$. So, finally,

$$\Delta E = - m (l(t_1)*\Delta l \dot \theta ^2)$$

But $E_1 = \frac 1 2 m l^2(t_1)* \dot \theta ^2 = E$ is the energy constant of the undisturbed oscillations. So:

$$\Delta E = - 2 \frac {\Delta l} l E $$

which is wrong.

 

[Orodruin]

Am I right ?

No. The derivative is a partial derivative wrt t. In order to integrate it like that it must be the total derivative.

 

[dRic2]

Oh right... what was I thinking!? Thank you very much.

But now I don't know how to solve that integral

 

[dRic2]

It has been 3 days since I started thinking on this problem. Still I have no clue how to solve that integral. I think there should be an easier way... Can someone provide any suggestion/insight ?

I would appreciate a lot.

 

[Orodruin]

Note the keyword that tells you the length changes ”slowly”. What does this tell you about the solution?

 

[dRic2]

Note the keyword that tells you the length changes ”slowly”. What does this tell you about the solution?

I think it means that the system undergoes a "reversible transformation", i.e. the pendulum goes from a state of equilibrium with energy $E_0$ to a state of equilibrium of energy $E_0+dE$. But I'm not sure. And even if it was true, I do not know how to exploit this information