# Uniform Rod Attached To Spring Motion Equation Problem

## Homework Statement

'Figure 2 shows a uniform rod of length L= 0.2m and mass m=0.2kg pivoted at one end. The other end is attached to a horizontal spring with spring constant k =3.0 N/m. The spring is neither stretched nor compressed when the rod is perfectly vertical. You can also assume that the force due to the spring is always horizontal.

a) Show that the equation of motion for the rod is:

$$\frac{d^2\theta}{dt^2}= \frac{3k}{m}\sin\theta\cos\theta - \frac{3g}{2L}\sin\theta$$

## Homework Equations

$$F=-kx, F=ma, F=-mg\sin\theta$$

## The Attempt at a Solution

I have no real idea of how to tackle this problem, I presume we need to resolve the system horizontally in terms of the restoring forces needed by both parts, which in this case would be:

$$F=-kx-mg\sin\theta$$

After that, I have no idea how to tackle the problem, if someone could help point me in the right direction, it would be much appreciated as I'm getting a little bit stressed out at not being able to get the grips with this question...

For linear motion we use F = ma.

Similarly for angular motion we use

T = I$\alpha$

where T = torque, I = moment of inertia and $\alpha$ is the angular acceleration.