Oscillation of a Rod: Finding Frequency

[Fibo112]

Homework Statement

As in picture b) a uniform rod is displaced by a small angle a, the question is to find its frequency.
(small angle approximations are allowed)

Homework Equations

Mtotal*a c.o.m=Fext

The Attempt at a Solution

I figure the only relevant external force is the komponent of gravity tangential to the center of masses path. This force is equal to mgsin(a) a beeing the angle of displacement from the vertical(of the center of mass). This seems to produce the standard differential equation that I would have for a point particle. Where is my mistake? My solution is not correct

 

[BvU]

Hi Fibo,

Is this for part (a) of the illegible exercise or for part (b) ?

 

[Fibo112]

Hi Fibo,

Is this for part (a) of the illegible exercise or for part (b) ?

Part b

 

[haruspex]

is displaced by a small angle a

Exactly how it is displaced is important. The diagram is not completely clear to me, so you may need to translate more of the text.
It looks like the rod is suspended horizontally at its midpoint, then the string is displaced from the vertical but in the plane containing both string and rod. Right so far?
When displaced, is the rod still horizontal, or is it still at right angles to the string? (Not saying this matters, just asking.)

 

[BvU]

In part (a) the rod only moves as a point mass. Not so in part (b)

 

[Fibo112]

The rod is always at a right angle with the string( it can be considered rigid with no mass in the string).

 

[haruspex]

The rod is always at a right angle with the string( it can be considered rigid with no mass in the string).

Just realized the question I should have asked: is it a string or is it a massless rod to which the massive rod is fixed at a right angle?

 

[Fibo112]

Just realized the question I should have asked: is it a string or is it a massless rod to which the massive rod is fixed at a right angle?

The latter.

 

[haruspex]

The latter.

Ok, so what is its moment of inertia about the axis?

 

[BvU]

Fibo, if you post a clear problem statement for (b), Haru will instantly see what's going on ...
The picture is unsharp and in a strange Sprache

 

[BvU]

Ah, we're on track

 

[Fibo112]

1/12mL^2+ mh^2 h beeing the length of the massless rod and L the length of the massive rod

 

[haruspex]

1/12mL^2+ mh^2 h beeing the length of the massless rod and L the length of the massive rod

Right, so is the differential equation exactly the same as for case a)?

 

[Fibo112]

No. I think I am starting to get it. I assumed since the massless rod was connected to the center of the massive rod it could only exert a force there, this seems to not be the case though?

 

[haruspex]

No. I think I am starting to get it. I assumed since the massless rod was connected to the center of the massive rod it could only exert a force there, this seems to not be the case though?

There will also be a torque.

 

[Fibo112]

Yes, so the massless rod must apply a force at a point other than the center right?

 

[haruspex]

Yes, so the massless rod must apply a force at a point other than the center right?

No, it applies a force and a torque at the centre.
Ok, in the real world you cannot apply a torque at a point. The connection between the two rods would have to have some small width, but we don't care about that - just treat it as a torque.

 

[Fibo112]

But if the force is applied at the center where does the angular momentum of the massive rod about its C.o.m comd from?

 

[haruspex]

But if the force is applied at the center where does the angular momentum of the massive rod about its C.o.m comd from?

It's easier to work in terms of moment of inertia and torque about the axis of rotation. What is that torque?

 

[Fibo112]

It will be mghsin(a), a being the angle with the vertical. But where does the massive rods angular momentum about its c.o.m come if the massless rods force acts on its center?

 

[haruspex]

It will be mghsin(a), a being the angle with the vertical. But where does the massive rods angular momentum about its c.o.m come if the massless rods force acts on its center?

As I posted, you can treat the two rods as a single system. You have a MoI and a torque for that system, producing an angular acceleration.
But to answer you question (if I understand it), the rotational inertia of the massive rod generates a torque on the massless rod, slowing its angular movement.

 

[Fibo112]

As I posted, you can treat the two rods as a single system. You have a MoI and a torque for that system, producing an angular acceleration.
But to answer you question (if I understand it), the rotational inertia of the massive rod generates a torque on the massless rod, slowing its angular movement.

But what is causing the torque on the massive rod about its center of mass?

 

[haruspex]

But what is causing the torque on the massive rod about its center of mass?

Action and reaction.
If it were freely jointed, the massive rod would not change orientation. It would swing as though a point mass. The angle between the rods would keep changing.
Making it a rigid joint means the change of angle is resisted. That creates a torque one way on the massive rod and the opposite way on the massless rod.

 

[Fibo112]

Ok, I think I understand it now. Just to confirm the massless rod will exert some force that is not exactly at the center of the massive rod?

 

[haruspex]

Ok, I think I understand it now. Just to confirm the massless rod will exert some force that is not exactly at the center of the massive rod?

You can think of it that way, or as a force at the centre plus a torque about the centre.

 

[Fibo112]

Ok thanks