Physics 218 Chapter 18

1. Physics 218Chapter 18 Prof. Rupak Mahapatra Physics 218, Lecture XXIV

2. Chapter 18: Periodic Motion • This time: • Oscillations and vibrations • Why do we care? • Equations of motion • Simplest example: Springs • Simple Harmonic Motion • Next time: • Energy } Concepts } The math Physics 218, Lecture XXIV

3. Physics 218, Lecture XXIV

4. What is an Oscillation? • The good news is that this is just a fancy term for stuff you already know. It’s an extension of rotational motion Stuff that just goes back and forth over and over again “Stuff that goes around and around” • Anything which is Periodic • Same as vibration • No new physics… Physics 218, Lecture XXIV

5. Examples Lots of stuff Vibrates or Oscillates: • Radio Waves • Guitar Strings • Atoms • Clocks, etc… In some sense, the Moon oscillates around the Earth Physics 218, Lecture XXIV

6. Why do we care? Lots of engineering problems are oscillation problems • Buildings vibrating in the wind • Motors vibrating when running • Solids vibrating when struck • Earthquakes Physics 218, Lecture XXIV

7. What’s Next • First we’ll “model” oscillations with a mass on a spring • You’ll see why we do this later • Then we’ll talk about what happens as a function of time • Then we’ll calculate the equation of motion using the math Physics 218, Lecture XXIV

8. Simplest Example: Springs k What happens if we attach a mass to a spring sitting on a table at it’s equilibrium point (I.e., x = 0) and let go? What happens if we attach a mass, then stretch the spring, and then let go? Physics 218, Lecture XXIV

9. Questions • What are the forces? Hooke’s Law: F= -kx • Does this equation describe our motion? x = x0 + v0t + ½at2 Physics 218, Lecture XXIV

10. The forces No force Force in –x direction Force in +x direction Physics 218, Lecture XXIV

11. More Detail Time Physics 218, Lecture XXIV

12. Some Terms Amplitude:Max distance Period:Time it takes to get back to here Physics 218, Lecture XXIV

13. Overview of the Motion • It will move back and forth on the table as the spring stretches and contracts • At the end points its velocity is zero • At the center its speed is a maximum Physics 218, Lecture XXIV

14. Simple Harmonic Motion Call this type of motion Simple Harmonic Motion (Kinda looks like a sine wave) Next: The equations of motion: Use SF = ma = -kx (Here comes the math. It’s important that you know how to reproduce what I’m going to do next) Physics 218, Lecture XXIV

15. Equation of Motion k A block of mass m is attached to a spring of constant k on a flat, frictionless surface What is the equation of motion? Physics 218, Lecture XXIV

16. Summary: Equation of Motion Mass m on a spring with spring constant k: x = A sin(wt + f) Where w2 = k/m A is the Amplitude • is the “phase” (phase just allows us to set t=0 when we want) Physics 218, Lecture XXIV

17. Simple Harmonic Motion At some level sinusoidal motion is the definition of Simple Harmonic Motion A system that undergoes simple harmonic motion is called a simple harmonic oscillator Physics 218, Lecture XXIV

18. Understanding Phase: Initial Conditions A block with mass m is attached to the end of a spring, with spring constant k. The spring is stretched a distance D and let go at t=0 • What is the position of the mass at all times? • Where does the maximum speed occur? • What is the maximum speed? Physics 218, Lecture XXIV

19. Check: This looks like a cosine. Makes sense… Spring and Mass Paper which tells us what happens as a function of time Physics 218, Lecture XXIV

20. Example: Spring with a Push We have a spring system • Spring constant: K • Mass: M • Initial position: X0 • Initial Velocity: V0 Find the position at all times Physics 218, Lecture XXIV

21. What is MOST IMPORTANT? Simple Harmonic Motion X= A sin(wt + f) • What is the amplitude? • What is the phase? • What is the angular frequency? • What is the velocity at the end points? • What is the velocity at the middle? Physics 218, Lecture XXIV