Horizontal mechanical oscillator We set a horizontal mechanical oscillator formed of a spring of negligible mass and of stiffness constant k=40N/m, having a fixed extremity while its other extremity is attached to a

solid (S) of center of gravity G and of mass m=100g. G oscillates along a horizontal axis x'ox of unit vector i. level of the gravitational energy. Take g=10m/s². The graph of fig.2 shows the variations of x as a function of time.

(S)

Fig.1

At an instant to-0 taken as origin of time, we start recording the variations of the abscissa x=OG, where O is the equilibrium position of G (fig. 1). The horizontal plane passing by G is considered as a reference

x(cm)

0,5

10 8

1

1.5

1. At an instant t, write down the expression of the mechanical energy ME of the system(oscillator, earth) as a function of m, k, x and the velocity V of G.

2. The mechanical energy of the system(oscillator, earth) is conserved, Justify; then referring to fig.2

show that ME= 0.2 J.

3. Referring to fig.2, determine at the instant to=0: 3.1. The value of the elastic potential energy of the oscillator,

3.2.The value of the kinetic energy of (S), 3.3. The direction of the motion of (S) and the algebraic value of the velocity Vo of (S).

4. 4.1. Establish the 2nd order differential equation that governs the motion of G.

4.2. The time equation of the motion of (S) is x-Xmsin(coot+p). Determine the constants 000, Xm and o.

4.3. Calculate the proper period To of the oscillator. 5. Specify the direction of the motion of (S) at the instant when the graph cuts the time axis for

the first time and determine the algebraic value of the velocity of (S) at the same instant.

6. Determine the velocity of (S) when x = 8 cm.

(s)

Fig.2