# What is the time taken for a flag pole to fall over and hit the ground?

1. Feb 19, 2012

### Charlie261

1. The problem statement, all variables and given/known data

The flag pole is 10m tall. The base of the pole has rusted and the pole topples over. What is the time taken for the pole to hit the ground?

s = distance = 10m
u = initial velocity = 0 m/s
v = final velocity = unknown
a = g = acceleration due to gravity = 9.81 m/s2
t = time taken

2. Relevant equations

s = ut + ½at2

3. The attempt at a solution

√2s/g = t = 1.43s

But this pole falling over is not in free fall. It is going to take longer. Please help.

2. Feb 19, 2012

### Staff: Mentor

What is causing the pole to fall? What forces act on a perfectly vertical pole?

3. Feb 20, 2012

### Charlie261

The base of the pole at ground level has completely rusted. One side of the pole at the base crumbles slightly just enough for the pole to not be verticle and so the pole is not balanced and the pole falls over due to gravity.

Another scenario would be that a person tried to balance a pole on its end. The pole is balanced and remains in the vertical for a few seconds and then loses balance and topples over. What would be the most time taken for a pole to move from the vertical to the horizontal due to the force of gravity alone?

Did not want to over complicate things but if it helps:

Assume a very small intial force at the top of the pole to make the pole not vertical.

ie. velocity of top of pole 1mm/s at a distance 1mm from the vertical position.

Assume there is no side ways movement at the base of the pole due to friction. ie rough ground.

Last edited: Feb 20, 2012
4. Feb 23, 2012

### Charlie261

Since nobody has come up with a solution I am beginning to think that there is no equation for an object falling over.
I was also hoping to work out the velocity that the top of the pole would hit the ground.

If someone could explain the difficulties of why there is no easy solution that would help me.

5. Feb 23, 2012

### Staff: Mentor

Oh, there are equations all right. You can use conservation of energy to determine how angular velocity depends upon the angle from vertical. Add in initial angular velocity to get the pole away from the vertical unstable equilibrium point and you end up with a differential equation to solve for angle versus time. I believe that this leads to some variety of elliptic integral to solve, which is not exactly "introductory material".
Well, that's an easy one. Conservation of energy (center of mass height change versus rotational velocity).