HELP QUICK Non-Uniform Circular Acceleration

[mcovalt]

I need a formula by tomorrow morning! Sorry for being annoying, but I am in way over my head.

For a calculus project I said I would do a video on the calculus in a physics problem, but I have no idea how to get the physics formula!

I need a formula to compute the time it takes for a uniform cylindrical beam to fall in a circular path from an upright position with a negligible push (think a pencil falling over after an attempt to balance it on its point).

I know how to compute angular velocity and all those torque questions, but due to the exponential increase of gravity's force while the rod is falling, I have no idea!

Help me out please!

 

[berkeman]

I need a formula by tomorrow morning! Sorry for being annoying, but I am in way over my head.

For a calculus project I said I would do a video on the calculus in a physics problem, but I have no idea how to get the physics formula!

I need a formula to compute the time it takes for a uniform cylindrical beam to fall in a circular path from an upright position with a negligible push (think a pencil falling over after an attempt to balance it on its point).

I know how to compute angular velocity and all those torque questions, but due to the exponential increase of gravity's force while the rod is falling, I have no idea!

Help me out please!

Hope you figured it out in time...

 

[mcovalt]

I thought I did, but I did not. Take a look at my work:

F = -mg
T = -mg (L/2) cos theta
I = (2/3) mL^2

T = I alpha
alpha = T / I = -mg (L/2) cos theta / (2/3) mL^2 = -(3g/L) cos theta

d^2 theta / dt^2 = alpha
d^2 theta / dt^2 = -(3g/L) cos theta

But I am stumped. Because I am taking a late grade on this, help would still be appreciated.

 

[mcovalt]

$\ \frac{d^2\theta}{dt^2}=\frac{g\sin\theta-\frac{L}{2}\frac{d\theta}{dt}\cos2\theta}{\frac{I_{G}}{mL/2}+\frac{L}{2}\sin2\theta}$