1. -Damped and Forces Oscillations -Resonance AP Physics C Mrs. Coyle

2. Objectives Identify and analyze forced and damped oscillations. Develop a qualitative understanding of resonance Identify situations in which a system will resonate in response to a sinusoidal external force.

3. Damped Oscillations Non conservative forces are present (ex:friction, resistive forces, damping forces by a “dashpot” device). The amplitude and thus the mechanical energy is reduced over time.

4. Solution: Damped Oscillation- Ex 1  F x = -k x – bv = ma Equation of Motion:

5. Damped Oscillation The amplitude is reduced over time and eventually the oscillation stops.

6. Damped Oscillation- Ex 2 Resistive (retarding force), R R = - b v where b is a constant called the damping coefficient  F x = -k x – bv x = ma x Equation of Motion: Solution viscous liquid

7. Types of Damping A.Underdamped If R max = bv max < kA B.Critically damped When b reaches a critical value b c such that b c / 2 m = k/m=  0 2, the system will not oscillate (quick return to equilibrium). is called the natural frequency C.Overdamped If R max = bv max > kA and b/2m >  0 (return to equilibrium without oscillation).

8. Video of dampers https://www.youtube.com/watch?v=xp2pGxF zrzI https://www.youtube.com/watch?v=xp2pGxF zrzI

9. Forced vibrations: when an external force causes a system to oscillate External Force  F x =

10. Amplitude of a forced (driven) oscillation: –  0 is the natural frequency of the undamped oscillator

11. How can a damped system have an undamped motion (no decrease in amplitude)? To compensate for the loss of mechanical energy due to the resistive force, apply a forced vibration of equal energy.

12. Resonance: increase in amplitude due to addition of an external force. When the frequency of the driving force is near the natural frequency (   ) an increase in amplitude occurs The natural frequency   is also called the resonance frequency of the system

13. At resonance the applied force and v are both proportional to sin (  t +  ), so force and velocity are in phase. The power transferred to the oscillator (P=F. v) is a maximum at resonance.

14. Takoma Narrows Bridge: collapses in November, 1940, under 42mph winds (opened in July 1940) https://www.youtube.com/watch?v=xox9BVS u7Ok https://www.youtube.com/watch?v=xox9BVS u7Ok