Bird sitting on a branch (vibrations and waves question)

[Odyssey]

Homework Statement

A bird lands near the tip of a branch and it is observed that initially it oscillates up and down, about once per second. Estimate how far the tip of the branch will be below its equilibrium position once the bird comes to rest.

Homework Equations

Newton's second law
Hooke's Law

The Attempt at a Solution

I am having trouble starting the question, and I guess that's drawing the free-body diagram of the bird.
I think the net force is in the y direction where F = T sin theta, where theta = arctan of y over L, y = vertical displacement from equilibrium and L is the length of the branch.

sin theta under small displacement is approximately tan theta which equals y/L.

This is the farthest I went. I guess it's still a long way to finding the position below equilibrium...

 

[andrevdh]

Use the spring formula for the period:

$$T = 2\pi \sqrt{\frac{m}{k}}$$

 

[chaoseverlasting]

Assuming the elasticity of the branch to be Y, we can say y=f*l/dl*A, where dl is the increase in length. The restoring force will be f=Y*A*dl/l. Now dl will vary with x, ie, dl=f'(x)dx as the branch is bending downwards.

At the same time, the weight of the branch opposes the restoring force and so does the weight of the bird. Therefore, if we assume the linear mass density of the branch to be constant, then if the total mass of the branch is M, then the mass of a small element dx at a distance x from the origin (which is at the point where the branch is connected to the tree), given by dm will be dm=M/L*dx.

Therefore, if you find the net torque acting on the branch about the origin due to the weight of the bird + branch and set it equal to the net torque due to the restoring force, you should have your answer. If you find the force using general expressions before the equilibrium is achieved, I think you should get an equation of SHM, but that is not required. Nice question.

 

[Odyssey]

Heh, thanks for the help! I got it! :)