PRIOR READING: Main 1.1, 2.1 Taylor 5.1, 5.2

1. SIMPLE HARMONIC MOTION:NEWTON’S LAW simple not simple PRIOR READING: Main 1.1, 2.1 Taylor 5.1, 5.2 http://www.myoops.org/twocw/mit/NR/rdonlyres/Physics/8-012Fall-2005/7CCE46AC-405D-4652-A724-64F831E70388/0/chp_physi_pndulm.jpg

2. q T m mg The simple pendulum Energy approach

3. q T m mg The simple pendulum Energy approach

4. q T m Newton Known torque mg This is NOT a restoring force proportional to displacement (Hooke’s law motion) in general, but IF we consider small motion, IT IS! Expand the sin series … The simple pendulum

5. L q T m mg The simple pendulum in the limit of small angular displacements What is (t) such that the above equation is obeyed?  is a variable that describes position t is a parameter that describes time "dot" and "double dot" mean differentiate w.r.t. time g, L are known constants, determined by the system.

6. REVIEW PENDULUM L q T m C, pare unknown (for now) constants, possibly complex mg Substitute: p is now known (but C is not!). Note that w0 is NOT a new quantity! It is just a rewriting of old ones - partly shorthand, but also "w" means "frequency" to physicists!

7. REVIEW PENDULUM L q T m mg Simple harmonic motion TWO possibilities …. general solution is the sum of the two and it must be real (all angles are real). If we force C' = C* (complex conjugate of C), then x(t) is real, and there are only 2 constants, Re[C], and Im[C]. A second order DEQ can determine only 2 arbitrary constants.

8. REVIEW PENDULUM L q T m mg Re[C], Im[C] chosen to fit initial conditions. Example: q(0) = 0 rad and dqdt(0) = 0.2 rad/sec

9. REVIEW PENDULUM L q T m mg

10. Remember, all these are equivalent forms. All of them have a known w0=(g/L)1/2, and all have 2 more undetermined constants that we find … how? Do you remember how the A, B, C, D constants are related? If not, go back and review until it becomes second nature!

11. L q T m mg The simple pendulum ("simple" here means a point mass; your lab deals with a plane pendulum) simple harmonic motion ( potential confusion!! A “simple” pendulum does not always execute “simple harmonic motion”; it does so only in the limit of small amplitude.) Period does not depend on qmax, f

12. Free, undamped oscillators – other examples k m q T No friction m k m mg x L I C q Common notation for all

13. • The following slides simply repeat the previous discussion, but now for a mass on a spring, and for a series LC circuit

14. REVIEW MASS ON IDEAL SPRING k Newton m Particular type of force. m, k known k m x Linear, 2nd order differential equation What is x(t) such that the above equation is obeyed? x is a variable that describes position t is a parameter that describes time "dot" and "double dot" mean differentiate w.r.t. time m, k are known constants

15. REVIEW MASS ON IDEAL SPRING k m k m C, pare unknown (for now) constants, possibly complex x Substitute: p is now known. Note that w0 is NOT a new quantity! It is just a rewriting of old ones - partly shorthand, but also “w” means “frequency” to physicists!

16. k m k m x Square and add: Divide: A, f chosen to fit initial conditions: x(0) = x0 and v(0) = v0

17. 2 arbitrary constants (A, f) because 2nd order linear differential equation

18. Position: • A, f are unknown constants - must be determined from initial conditions • w0, in principle, is known and is a characteristic of the physical system Velocity: Acceleration: This type of pure sinusoidal motion with a singlefrequency is called SIMPLEHARMONIC MOTION

19. THE LC CIRCUIT DIFFERENTIAL EQUATION L I C q Kirchoff’s law (not Newton this time) Same differential equation as the SHO spring!

20. What is inductance?? It is how much magnetic flux is created in the inductor coil by a given current I, in a wire. What is the voltage change across an inductor?A voltage change occurs WHEN there is change in magnetic flux (i.e. some of the energy is ‘converted’ to a magnetic field)

21. THE LC CIRCUIT L Kirchoff’s law (not Newton this time) I C q