Harmonic Oscillation

1. Harmonic Oscillation

2. Umberto Eco Foucault’s Pendulum “That was when I saw the Pendulum. The sphere, hanging from a long wire set into the ceiling of the choir, swayed back and forth with isochronal majesty. I knew-but anyone could have sensed it in the magic of that serene breathing-that the period was governed by the square root of the length of the wire and by , that number which, however irrational to sublunar minds, through a higher rationality binds the circumference and diameter of all possible circles. The time it took the sphere to swing from end to end was determined by the arcane conspiracy between the most timeless of measures: the singularity of the point of suspension, the duality of the plane’s dimensions, the triadic beginning of , the secret quadratic nature of the root, and the unnumbered perfection of the circle itself. “ This presentation is easier to understand than Eco’s book!

3. If a force F acts on a spring, the length x changes. The change is proportional to the restoring force (Hooke’s Law). A spring force develops that is opposite and equal to the outside force. Thus we obtain F = -kx for the restoring force. The proportionality constant k is called the spring constant and depends on the measurements and natural characteristics of the spring. If the spring is hung vertically and a mass is attached, then the acceleration of gravity leads to the gravitational force Fg = mg and we can write F = Fg - kx = mg

4. A body with a mass m hangs from a spring. If the mass is disturbed from rest, it experiences an acceleration due to the restoring force: This takes effect in the direction of the equilibrium position. If the friction is negligible, then This results in the differential equation of the oscillator:

5. The solution of this differential equation is a function x(t) which describes the time dependence of the oscillating system’s movement This results in the place of a body to the time t at that moment. The maximal displacement from the resting position x=0 is called the amplitude x0 The argument of the sine wave (! t -  ) is called the phase. The starting position x0 at t = 0 is a result of the displacement in phase . The frequency and the angular frequency are

6. Problems 1.) You have a spring for which the spring constant is 100 N/m and you want to stretch it by 3.0 m. What force do you need to apply? F = - k ¢x = -300 N 2,) A weight on a spring is causing that spring to oscillate up and down. If the amplitude of the motion is 2 m and the angular frequency is 2.0 radians per second, where will the oscillating mass be after 20 seconds? 3.) A weight on a spring is oscillating with an angular frequency of 1.7 radians/sec and an amplitude of 0.7 m. What is the acceleration at 10 s?

7. Electronic Harmonic Oscillator An LC circuit is a resonant circuit that consists of an inductor represented by the letter L and a capacitor represented by the letter C. For this LC circuit the energy between the magnetic field of the inductor and the electrical field of the capacitor is periodically oscillated with the frequency of According to Kirchhoff’s law, the sum of the voltages in the circuit must be equal to zero. With the voltage over the inductor and the voltage over the capacitor

8. We obtain We know that Using some algebraic conversions we obtain with we obtain

9. Finally, dividing by LC results in a homogenous differential equation of second order We know the solution from our discussion of the mechanical oscillator: Inserting the cosine solution into the differential equation , we obtain

10. and thus :

11. Problems A 1 F capacitor is charged to 20 V and then is attached to a 6 H inductor. 1.) Find the frequency of the circuit’s oscillation.

12. 2.) Find the charge of the capacitor and current of the inductor at t = 5s. [Coulomb] = V F = V (As/V) = As