2. Physics 205

3. Phys 205: Vibrations and Waves Main Ideas zHooke’s Law zEnergy Conservation applied to a spring zSpring as an exemplar of other systems zSimple Harmonic Motion zDamped versus Free Motion

4. Phys 205: Vibrations and Waves Hooke’s Law becomes: The function x(t) is the displacement-time function for the spring-mass oscillator

5. Phys 205: Vibrations and Waves zOur intuition tells us that x(t) looks like: Which can be described by the equation... How can we show that this really is so? amplitude angular frequency

6. Phys 205: Vibrations and Waves Feed into How can we get the left and right hand sides of the equation to match?

7. Phys 205: Vibrations and Waves “grind out the derivatives and equate both sides”

8. Phys 205: Vibrations and Waves Velocity and acceleration zHow can we derive the following?

9. Phys 205: Vibrations and Waves Velocity and acceleration

10. Phys 205: Vibrations and Waves The Simple (and not so simple) Pendulum zA pendulum can, under appropriate circumstances act as a simple harmonic oscillator zThe sine component of W provides a restoring torque

11. Phys 205: Vibrations and Waves This can be expressed... zRecall zbut, if L = length of string then:

12. Phys 205: Vibrations and Waves zRe-arrange to give: zNow, notice the similarity with

13. Phys 205: Vibrations and Waves Important step... zIf the angle is small - less than about 0.1 radians (7 degrees) then z We can re-write to get:

14. Phys 205: Vibrations and Waves From this we can zFind an expression for the period of a simple pendulum... Try it!

15. Phys 205: Vibrations and Waves General equation for SHM... zWhenever the acceleration in the system is directly proportional and opposed to the displacement, SHM will result: Acceleration = - (Displacement) Some constant “opposed to”