1. Chapter 15 Oscillatory Motion

2. Recall the Spring Since F=ma, this can be rewritten as: Negative because it is a restoring force. In other words, if x is positive, the force and acceleration is negative and vice versa. This makes the object oscillate and therefore, it undergoes simple harmonic motion.

3. Now pay attention Remember that: can be rewritten as:So

4. To make this differential equation easier to solve, we make then After solving, we get: Note: This requires knowledge on how to solve differential equations. It is more important to know what the solution is.

5. Simple Harmonic Motion Equation for distance (x) as time (t) changes. General formula for simple harmonic motion. A, ω, ϕ are constants A is Amplitude - For springs: max value of distance (x) (positive or negative) - Maximum value the wave alternates back and forth between ω is Angular frequency → - How rapidly oscillations occur - Units are rad/s Remember:

6. is the phase constant is the phaseEssentially is the shifts of the wave These determine the starting position of the wave

7. General Concepts Anything with behaviors which have formulas that look like these are undergoing simple harmonic motion and can be measured using the same method as the spring.

8. Period and frequency Period: time for 1 full oscillation frequency: number of oscillations per second Measured in cycles per second - Hertz (Hz) For springs Note: frequency (f) and angular frequency (ω) measure the same thing but with different units. They differ by a factor of 2 pi.

9. Velocity and Acceleration Velocity of oscillation Acceleration of oscillation Note: magnitude of maximum values are when the sin and cosine arguments equal 1

10. Energy of Simple Harmonic Oscillators Remember that: After substituting the equations of velocity(v) and distance(x) for simple harmonic oscillations, we get:

11. Applications: Simple Pendulum mg T The restoring force for a pendulum is where which is the arclength or the path the ball travels along thus

12. Simple Pendulums continued Notice thatalmost looks like According to the small angle approximation, which states that sinθ ≈ θ if θ is small (about less than 10°) We can rewrite the equation to be which is exactly in the form for simple harmonic motions whereso then we can now use all the other formulas for simple harmonic motions for the case of a pendulum for small angles

13. Applications: Torsional Pendulum When a torsion pendulum is twisted, there exists a restoring torque which is equal to: This looks just likebut in rotational form Thus, we can apply what we know about angular motion to get information about this object’s simple harmonic oscillations

14. Torsion Pendulum continued Remember: After substitutingwe get whereand