Small Oscillations around equilibrium

[LiorE]

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

Homework 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:

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

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?

 

[LiorE]

Correction:

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

 

[LiorE]

Anyone?

 

[LiorE]

Seriously, can no one say anything?

 

[Redbelly98]

Just saw your question.

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?

You're almost there.

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