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## Homework Statement

A pencil of length l = 0.2 m is balanced on its point. How much time does it take to fall? Assume that the pencil is a massless rod and all of its mass is at the tip. To make the math easier, assume the small angle approximation.

## Homework Equations

theta (double dot) = g/l sin theta

T = 2 pi (l/g) ^ (1/2)

## The Attempt at a Solution

Since the pencil seems to be an inverted pendulum and the fall would be only 1/4 of the period, I first tried T/4 = pi/2 (.2/9.8) ^ (1/2). But that is not a correct solution.

Next I tried to rearrange and integrate

1/sin theta d

^{2}theta = g/l dt

^{2}

1/sin theta = sin theta/ sin

^{2}theta= sin theta/(1 - cos

^{2}theta)

let u = cos theta so du=- sin theta d theta

Integral( 1/ u

^{2}- 1) du = 1/2 ln(u - 1) - 1/2 ln (u + 1)

[ 1/2 ln (cos theta -1) - 1/2 ln (cos theta + 1)] d theta = gt/l dt

I get lost here because the cos pi/2 is zero which makes the above meaningless.

The way I interpret the problem, all the mass is at the balancing point. So how can you use energy to calculate potential difference or KE?

If someone could just give a hint as to the approach, I would be greatful.

Thank you.