(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A mass m is suspended by a massless string of varying length l = l_{0}- vt, where v is constant. The mass is released at angle [\theta]_{0}from rest.

(a) Write down the Lagrangian and find the equation of motion

(b) Show that these equations reduce to those of a simple pendulum for the case v -> 0

(c) (Now this is the hard part) Solve for the approximate motion for small amplitude. The amplitude at time t = 0 is [itex]\theta[/itex]_{0}. Qualitatively describe the motion.

2. Relevant equations

U = -mgy = -mg(l_{0}-vt)cos([itex]\theta[/itex])

T = 1/2mV^{2}= (1/2)mv^{2}+ (1/2)m[itex]\ddot{\theta}[/itex]^{2}(l_{0}- vt)^{2}

3. The attempt at a solution

(a) Using L=T-U, the Lagrangian is given by:

L = 1/2mVThe equation of motion is found by using the Euler-Lagrange equation. I found that^{2}= (1/2)mv^{2}+ (1/2)m[itex]\ddot{\theta}[/itex]^{2}(l_{0}- vt)^{2}+ mg(l_{0}-vt)cos([itex]\theta[/itex])

[itex]\ddot{\theta}[/itex] = -gsin[itex]\theta[/itex] / (l_{0}- vt)

(b) This one is just painfully obvious. I think it was only asked as a check for the students.

(c) This is the part I can't seem to get. I know that in the small angle approximation sin[itex]\theta[/itex] [itex]\rightarrow[/itex] [itex]\theta[/itex]. My intuition tells me that I should end up with a periodic function with an amplitude that decreases with time and some sort of time dependence such that the frequency of oscillation increases with time . As the string becomes shorter the mass cannot swing as far. The period of a simple pendulum in proportional to [itex]\sqrt{l}[/itex]. Since l is decreasing at a constant rate, so does the period. So I expect to have a cos((l_{0}- vt)[itex]\theta[/itex]) term. Or something similar with an exponential. It's just solving the differential equation that has got me stuck.

Any suggestions will be appreciated. Thank you for your help!

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# Homework Help: Pendulum with time dependent length

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