A rotating rod acted upon by a perpendicular force

[Hamiltonian]

Homework Statement

A uniform rod of mass M and length L pivoted at its center of mass is acted upon by a force F perpendicular to the line passing through the center of the rod. (the force F remains perpendicular always to the line passing through the center of the rod) find the time taken by the rod to cover an angle theta?

Relevant Equations

-

$$\tau = I\alpha$$
$$FL/2 = I\omega^2L/2$$
$$T = 1/\theta \sqrt{F/I}$$

would this be correct?
I came up with this more basic question to solve a slightly harder question so I do not know the answer to the above-stated problem.

 

[haruspex]

$$\tau = I\alpha$$

Yes.

$$FL/2 = I\omega^2L/2$$

How did you get this? It is dimensionally inconsistent.

$$T = 1/\theta \sqrt{F/I}$$

Does it make sense that increasing theta reduces the time?

 

[Hamiltonian]

How did you get this? It is dimensionally inconsistent.

disclaimer I am new to rigid body dynamics

i thought the $\tau$ = FL/2 = I$\alpha$ = I$\omega^2$(L/2)
from this I got $\sqrt{F/I} = \omega$
my final answer is $$ t = \theta\sqrt{I/F}$$
not what I stated above

 

[haruspex]

$I\alpha = I\omega^2(L/2)$

You seem to have a basic misunderstanding there. What principle are you using?

 

[Hamiltonian]

You seem to have a basic misunderstanding there. What principle are you using?

ok I think I get what I did wrong there.
$$I\alpha = Id(\omega)/dt$$ not what is wrote above

 

[haruspex]

ok I think I get what I did wrong there.
$$I\alpha = Id(\omega)/dt$$ not what is wrote above

Ok, so integrate that, knowing alpha is constant.
Btw, the question omits to say where along the rod the force acts. I see you are taking it to be at the tip.

 

[Hamiltonian]

Ok, so integrate that, knowing alpha is constant.
Btw, the question omits to say where along the rod the force acts. I see you are taking it to be at the tip.

$$\omega = FLt/2I$$

$$t = 2\sqrt{I\theta/FL}$$

 

[Hamiltonian]

Right, but you won’t need the second equation.
So how to get from there to an equation involving theta?

we need to find t($\theta$)
and $$\omega = d\theta/dt$$
so u can just replace $\omega$ with what I have stated above

 

[Hamiltonian]

$$t = 2\sqrt{I\theta/FL}$$

 

[Hamiltonian]

thanks for all the help :D

 

[haruspex]

$$t = 2\sqrt{I\theta/FL}$$

Sorry about my previous post... I misread yours. I've deleted it.