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Exercise 1 (10 points): We consider the mechanical oscillatory system of the Figure 1. The rigid \( \operatorname{rod} \mathrm{AB} \) is of length \( 2 \ell(\mathrm{OA}=\mathrm{OB}=\ell) \) and a negligible rrass. It carries to its extremates the punctual masses \( m \) and \( M \). The two identical springs are welded at a point \( O \) ' to the \( \operatorname{rod}\left(\mathrm{OO}^{\prime}=\mathrm{a}\right) \). At rest \( (\theta=0) \), the rod is vertical and the springs are not deformed. The rod is turned through an angle \( \theta \) (small oscillations) around the rotation axis \( O \). 1) Determine the kinetic and potential energies of the system as a function of \( \theta \). 2) Used the Lagrange method to establish the differential equation of the motion. 3) Find the solution \( \theta(t) \). Figure 1 4) For \( m=M \) and \( a=\ell / 2 \), deduce the period ( \( T \) ) of the oscillations.

          Exercise 1 (10 points):
We consider the mechanical oscillatory system of the Figure 1. The rigid \( \operatorname{rod} \mathrm{AB} \) is of length \( 2 \ell(\mathrm{OA}=\mathrm{OB}=\ell) \) and a negligible rrass. It carries to its extremates the punctual masses \( m \) and \( M \). The two identical springs are welded at a point \( O \) ' to the \( \operatorname{rod}\left(\mathrm{OO}^{\prime}=\mathrm{a}\right) \). At rest \( (\theta=0) \), the rod is vertical and the springs are not deformed. The rod is turned through an angle \( \theta \) (small oscillations) around the rotation axis \( O \).
1) Determine the kinetic and potential energies of the system as a function of \( \theta \).
2) Used the Lagrange method to establish the differential equation of the motion.
3) Find the solution \( \theta(t) \).

Figure 1
4) For \( m=M \) and \( a=\ell / 2 \), deduce the period ( \( T \) ) of the oscillations.

Exercise 1 (10 points):
We consider the mechanical oscillatory system of the Figure 1. The rigid rodAB is of length 2 ℓ(OA=OB=ℓ) and a negligible rrass. It carries to its extremates the punctual masses m and M. The two identical springs are welded at a point O ' to the rod(OO^'=a). At rest (θ=0), the rod is vertical and the springs are not deformed. The rod is turned through an angle θ (small oscillations) around the rotation axis O.
1) Determine the kinetic and potential energies of the system as a function of θ.
2) Used the Lagrange method to establish the differential equation of the motion.
3) Find the solution θ(t).

Figure 1
4) For m=M and a=ℓ / 2, deduce the period ( T ) of the oscillations.

Added by Aaron M.

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