Oscillations with small amplitude

Let me check I'm getting the right answer first. I make the initial d.e.

$\ddot{\theta}=-\frac{\sqrt{3}}{2}\theta$

Is that correct?

$T= 2 \pi \sqrt{\frac{2l}{\sqrt{3} g}}$

So, nope, unfortunately.

OK, couple of slips (rather rusty), but the basic idea was correct.

If you consider a displacment of theta from the horizontal for BC, then:
Angle to C from the vertical at A is 30+theta (assuming C is above B), and angle of B from the vertical at A is 30-theta.

Then considering moments about A, the total moment is:

$mgl(\sin(30+\theta) - \sin(30-\theta))$

Expand the sines and simplify, and equate to "appropriate angular acceleration"

Edit:

Quick check, that should be

Expanding the brackets I get $\sqrt{3} mgl \theta$, letting theta be small, what after this? I've never tackled a problem like this with moments.

...

As you wouldn't do it that way, you could post your methodology.

I will be away for a few hours though.

As per the edit in my previous post.

Where does the l^2 come from? I'm assuming you're using transverse acceleration as r d^2theta/dt^2, but I can't seem to get r=l^2.

But we've worked out the moment about A, what connection does it have to the transcerse acceleration? (Never seen this before)


wait. I see it now I think

angular acceleration = l d^2theta/dt^2
hence the moment about A is l^2 d^2theta/dt^2, then equate the two.

1) are there any other ways of tackling the problem?
2) how would you do thrust, like what would you consider?

The "moment" about a point is the torque on the system about that point. The torque is related to the angular acceleration via the moment of inertia:

$\mathbf{\tau}_A = I_A\ddot{\mathbf{\theta}}$

Since $I_A = \Sigma_i m_i r_i^2 = ml^2 + ml^2 = 2ml^2$ then you can write $2ml^2 \ddot{\mathbf{\theta}} = -mgl\sin{(30 + \theta)} + mgl\sin{(30 - \theta)}$


It is also possible to solve this system through energy considerations, since the total energy is a conserved constant.

1) are there any other ways of tackling the problem?
2) how would you do thrust, like what would you consider?

You could consider each mass separately and work out the transverse acceleration, using F=ma. This would include the thrust in the bar. It seems rather fiddly getting the angles right.

But you will have two equations and can eliminate T to get the d.e. and hence the period.

You can also, equate the two, and get T as a function of theta, m ,and g.

Going to be honest, not seen thrust in any question before.

So, does it act at both ends affecting both particles, hence to find the total thrust you just sum the two?

Going to be honest, not seen thrust in any question before.

So, does it act at both ends affecting both particles, hence to find the total thrust you just sum the two?