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Small Oscillations around equilibrium

  • Thread starter LiorE
  • Start date
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1. Homework Statement

The problem is: A point pendulum is being accelerated at a constant acceleration of a. Basically what's required is to find the equations of motion, the equilibrium point, and to show that the frequency of small oscillations about the e.p. is: [tex]\omega=L^{-1/2} (a^2+g^2)^{-1/4}[/tex]

2. Homework Equations

The equation of motion I've arrived at is:

[tex] \ddot{\theta} + \frac{a\cos\theta + g\sin\theta}{L} = 0[/tex]

So the e.p is:

[tex]\theta_0 = -\arctan(a/g)[/tex]

3. The Attempt at a Solution

If we do a Taylor expansion for [tex]\eta << 1[/tex] around [tex]\theta_0[/tex]:

[tex]\cos(\theta_0+\eta) = \cos\theta_0 - \eta \sin\theta_0+\ldots[/tex]
[tex]\sin(\theta_0+\eta) = \sin\theta_0 + \eta \cos\theta_0+\ldots[/tex]

We end up with:

[tex]\ddot{\eta} + \frac{\cos\theta_0}{L}\frac{g^2+a^2}{g} \eta = 0[/tex]

Then what am I missing here?
 

Attachments

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Correction:

[tex]
\omega=L^{-1/2} (a^2+g^2)^{1/4}
[/tex]
 
38
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Anyone?
 
38
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Seriously, can no one say anything?
 

Redbelly98

Staff Emeritus
Science Advisor
Homework Helper
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Just saw your question.

We end up with:

[tex]\ddot{\eta} + \frac{\cos\theta_0}{L}\frac{g^2+a^2}{g} \eta = 0[/tex]

Then what am I missing here?
You're almost there.

Express θ0 in terms of a and g, using the arctan relation you came up with earlier. Things will simplify.
 

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