# Small Oscillations around equilibrium

1. Mar 8, 2009

### LiorE

1. The problem statement, all variables and given/known data

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: $$\omega=L^{-1/2} (a^2+g^2)^{-1/4}$$

2. Relevant equations

The equation of motion I've arrived at is:

$$\ddot{\theta} + \frac{a\cos\theta + g\sin\theta}{L} = 0$$

So the e.p is:

$$\theta_0 = -\arctan(a/g)$$

3. The attempt at a solution

If we do a Taylor expansion for $$\eta << 1$$ around $$\theta_0$$:

$$\cos(\theta_0+\eta) = \cos\theta_0 - \eta \sin\theta_0+\ldots$$
$$\sin(\theta_0+\eta) = \sin\theta_0 + \eta \cos\theta_0+\ldots$$

We end up with:

$$\ddot{\eta} + \frac{\cos\theta_0}{L}\frac{g^2+a^2}{g} \eta = 0$$

Then what am I missing here?

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2. Mar 8, 2009

### LiorE

Correction:

$$\omega=L^{-1/2} (a^2+g^2)^{1/4}$$

3. Mar 9, 2009

Anyone?

4. Mar 9, 2009

### LiorE

Seriously, can no one say anything?

5. Mar 10, 2009

### Redbelly98

Staff Emeritus