# Problem in Classical Mechanics

## Homework Statement

I am stuck over a classical mechanics problem. I tried to solve it, but after having solved the first point, I got stuck on the second one. Here is the problem:

>A mechanical structure is composed by 4 rigid thin bars of length $\ell = 8\ m$, mass $m = 5\ kg$ each one. Those bars are connected each other at their extremities through frictionless hinges, such that the whole structure forms a vertical rhombus.
The structure is hanged from the ceiling by means of the upper hinge.
In the lower hinge, a mass $M = 30\ kg$ is hanged.
Between the two horizontal hinges, a rectilinear spring (whose mass is negligible) acts with elastic constant $k = 40.5\ N/m$ and rest length $L = 9\ m$, such that the angle between the two superior bars is $\theta = 40^{\circ}$.

>Calculate:

>1) The Force the spring exerts over one of the hinges, and the total tension of the bars connected to the spring in that point;

>2) What should be the elastic constant if $\theta$ remains the same, when the whole structure rotates with constant angular velocity $\omega = 2$ rad/s, around the vertical axis?

>3) The moment of Inertia with respect to the rotation axis of the previous question, and the angular momentum when it rotates with a general $\omega$.

Here is the figure

For the first point, I calculated, though some trigonometry, the length of the spring in those conditions. Calling $l$ the semi length, I got

$$l = \ell\sin\frac{\theta}{2} = 2.736\ m$$

So the total length is $2l = 5.472\ m$. Now the elastic force is simply

$$F_k = k\Delta l = k (L - 2l) = 142.88\ N$$

And it's ok.

For the second point of the first request I used

$$T_{\text{tot}} = 2T\cos 70 ~~~~~~~ 2Tcos 70 - F_k = 0$$

$$T = \frac{F_k}{2\cos 70} = 209\ N$$

And it's ok.

**Now I'm stuck over the second request**

**Numerical results:**

2) $$\Delta k = -15.5\ N/m$$

3) $$I = 49.9\ kg\cdot m^2$$ and $$L = 99.8\ kg\cdot m^2/s$$

Any help?

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