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I would like to preface this by saying that I solved the homework problem, but my professor gave me an added challenge of finding the period of the motion described in this problem.

The pendulum bob of mass

I don't have an exact diagram so here's a replica with all necessary labels:

After setting up the Lagrangian and finding the equations of motion (which have been checked by my professor), we get

$$m\ddot{x} + mr\ddot{\theta}\cos{\theta} - mr{\dot{\theta}}^2\sin{\theta} + 2kx = 0$$

$$\ddot{x} + r\ddot{\theta} + g\sin{\theta} = 0$$

First of all, we are assuming theta is very small. So far we have used an integral involving the mechanical and potential energy of a system to find the time it takes to travel from one position to another. The integral is essentially ##dt## written in terms of ##dx, E, V(x).## The problem is, I don't know what the maximum value of theta in this oscillation is. My professor thinks there is a way to make the period pop out of the equations under the small-angle approximations. After these approximations, I get

$$m\ddot{x} + mr\ddot{\theta} - mr{\dot{\theta}}^2\theta + 2kx = 0$$

$$\ddot{x} + r\ddot{\theta} + g\theta = 0$$

What immediately stood out to me was that both the ##x, \theta## accelerations could be canceled out in one step, giving ##2kx = m\theta(g + r{\dot{\theta}}^2)##. As I am writing this, I realize that separating the variables could be used here, and even better, that x can be written as an oscillator (duh).

So now I'm here:

$$\dot{\theta} = \sqrt{\frac{f(t)}{\theta} + C}$$ for some function of time ##f(t)## and some constant ##C##.

I don't know how to proceed, unless there is a way to solve this after finding

Thanks.

1. Homework Statement1. Homework Statement

The pendulum bob of mass

*m*shown in the figure below is suspended by an in-extensible string from point*p*. This point is free to move along a horizontal line under the action of the springs, each having an elastic constant*k*. Assume that point*p*is displaced a distance*x*to the right and then released; determine the equations of motion of the system.I don't have an exact diagram so here's a replica with all necessary labels:

**2. The Equations of Motion**After setting up the Lagrangian and finding the equations of motion (which have been checked by my professor), we get

$$m\ddot{x} + mr\ddot{\theta}\cos{\theta} - mr{\dot{\theta}}^2\sin{\theta} + 2kx = 0$$

$$\ddot{x} + r\ddot{\theta} + g\sin{\theta} = 0$$

## The Attempt at a Solution

First of all, we are assuming theta is very small. So far we have used an integral involving the mechanical and potential energy of a system to find the time it takes to travel from one position to another. The integral is essentially ##dt## written in terms of ##dx, E, V(x).## The problem is, I don't know what the maximum value of theta in this oscillation is. My professor thinks there is a way to make the period pop out of the equations under the small-angle approximations. After these approximations, I get

$$m\ddot{x} + mr\ddot{\theta} - mr{\dot{\theta}}^2\theta + 2kx = 0$$

$$\ddot{x} + r\ddot{\theta} + g\theta = 0$$

What immediately stood out to me was that both the ##x, \theta## accelerations could be canceled out in one step, giving ##2kx = m\theta(g + r{\dot{\theta}}^2)##. As I am writing this, I realize that separating the variables could be used here, and even better, that x can be written as an oscillator (duh).

So now I'm here:

$$\dot{\theta} = \sqrt{\frac{f(t)}{\theta} + C}$$ for some function of time ##f(t)## and some constant ##C##.

I don't know how to proceed, unless there is a way to solve this after finding

*f.*Is there a more general way to do this given the equations of motion?Thanks.