HELP QUICK Non-Uniform Circular Acceleration

AI Thread Summary
A formula is needed to calculate the time it takes for a uniform cylindrical beam to fall from an upright position due to gravity. The discussion highlights the challenge of accounting for the exponential increase in gravitational force during the fall. Key equations involving torque and angular acceleration are provided, but the user remains confused about the application. The user expresses urgency in needing the formula for a calculus project due soon. Assistance is requested to clarify the calculations and derive the necessary formula.
mcovalt
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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!
 
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mcovalt said:
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...
 
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.
 
\ \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}
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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