- #1

ttzhou

- 30

- 0

## Homework Statement

Suppose that there is a negligibly thin tree in the forest of infinite length that begun tipping over. Negating frictional effects from the pivoting, does the tree ever hit the ground?

## Homework Equations

My approach was to solve the problem for a tree of length [itex]l[/itex] and see what happens in the limit as [itex]l \rightarrow \infty[/itex]

[itex]

\tau = I\alpha

[/itex]

[itex]

\tau = Mg\cos{\theta}(l/2)\

[/itex]

[itex]

I = \frac{1}{3} Ml^2\

[/itex]

## The Attempt at a Solution

[itex]

I\alpha = Mg\cos{\theta} (l/2)

[/itex]

[itex]

\alpha = \frac{3gl}{2} \cos{\theta}

[/itex]

Now, I know how to approach and get a function for [itex]\omega[/itex] by using energy conservation; however, out of curiosity, would I be able to solve this DE purely mathematically?

In any case, [itex]\omega = \sqrt{\frac{3g(1-\sin{\theta})}{l}} [/itex]

I interpret this to mean that if I take the limit as the length goes to infinity, the tree simply has no angular speed and thus cannot fall over at all. However, this seems unsatisfactory as an answer - is it possible to actually derive an explicit function for [itex]\theta(t)[/itex]?

Thank you all for any help, and I apologize if I have made any sort of oversight or foolish mistake.