# Mass on a stick model.

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## Main Question or Discussion Point

I have need of a mathematical model of a mass at the end of a stick, pivoting about the opposite end. What I have done so far is not giving realistic results, would appreciate it if some one could point out my error.

Basic model :

$$\tau = I \ddot {\theta}$$

$$I = m r^2$$

The system is being pushed by an air cylinder (force F) with the fixed end mounted a distance L directly below the pivot, the moving end of the cylinder is mounted a distance r along the pivot arm from the pivot point. With $\phi$ the angle between the cylinder and the pivot arm. So I have :

$$\tau = r F sin( \phi)$$

My variable of interest will be the angle between the pivot arm and the line defined by the pivot and the fixed end of the cylinder, call this angle $\theta$

By the law of sines I get

$$\frac {Sin(\phi)} {L} = \frac {sin(\theta)} {x}$$
where x is the length of the cylinder.

I get x in terms of $\theta$ from the law of cosines

$$x^2 = L^2 + r^2 -2lr cos(\theta)$$

The differential equation is:

$$\ddot{ \theta } = \frac { \tau } {I}$$

See the attachment for a diagram.

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