Plane pendulum in terms of the x axis

In summary, the free motion of a plane pendulum with a non-small amplitude can be approximated by the expression \ddot{x}+w^{2}_{0} (1+x^{2}_{0}/L^{2})x - Ex^{3}=0, where w^{2}_{0}=g/L, E=3g/(2L^{3}), and L is the length of the suspension. This is derived from a second order non-linear equation \ddot{\Theta}+w^{2}_{0}sin^{2}\Theta=0, and can be solved by finding theta'' in terms of x'' and substituting it into the equation.
  • #1
oswaler
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Homework Statement



consider the free motion of a plane pendulum whose amplitude is not small. Show that the horizontal component of the motion may be represented by the approximate expression ( components through the 3rd order are included)

[tex]\ddot{x}[/tex]+w[tex]^{2}_{0}[/tex] (1+x[tex]^{2}_{0}[/tex]/L[tex]^{2}[/tex])x - Ex[tex]^{3}[/tex]=0

where w[tex]^{2}_{0}[/tex]=g/L , E=3g/(2L[tex]^{3}[/tex]) and L is the length of the suspension.

Homework Equations



we have a second order non-linear equation: [tex]\ddot{\Theta}[/tex]+w[tex]^{2}_{0}[/tex]sin[tex]^{2}[/tex][tex]\Theta[/tex]=0

The Attempt at a Solution


I'm pretty stuck on this. It seems that I need to get a function for x in terms of [tex]\Theta[/tex], but I'm not quite sure where to start. I see here that I'm not going to be able to use the small angle approximation sin[tex]\Theta[/tex]=[tex]\Theta[/tex].

Thanks for any help.
 
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  • #2
x = L(sin theta) => theta = arcsin x/L.

Try to find theta'' in terms of x'', and at some point sub in the value of theta'' from the last eqn, so as to get w0^2.
 

Related to Plane pendulum in terms of the x axis

1. What is a plane pendulum in terms of the x axis?

A plane pendulum in terms of the x axis refers to a type of pendulum that moves in a two-dimensional plane, with the x axis representing the horizontal direction of motion.

2. How does the length of the pendulum affect its motion in terms of the x axis?

The length of the pendulum affects its motion in terms of the x axis by changing the period of the pendulum's oscillation. A longer pendulum has a longer period, meaning it takes longer to complete one full swing.

3. What is the equation for the period of a plane pendulum in terms of the x axis?

The equation for the period of a plane pendulum in terms of the x axis is T=2π√(L/g), where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity.

4. How does the angle of release affect the motion of a plane pendulum in terms of the x axis?

The angle of release affects the motion of a plane pendulum in terms of the x axis by changing the amplitude of the pendulum's oscillation. A larger angle of release results in a larger amplitude and a longer period.

5. How does the mass of the pendulum affect its motion in terms of the x axis?

The mass of the pendulum does not affect its motion in terms of the x axis. The period of a pendulum is only dependent on the length and acceleration due to gravity, not the mass of the pendulum.

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