SIMPLE HARMONIC MOTION: NEWTON S LAW

Transcription

1 SIMPLE HARMONIC MOTION: NEWTON S LAW siple not siple PRIOR READING: Main 1.1, 2.1 Taylor 5.1,

2 Reeber, all these are equivalent fors. All of the have a nown 0 =(g/l) 1/2, and all have 2 ore undeterined constants that we find how? (t) Acos 0 t (t) B p cos 0 t B q sin 0 t (t) C expi 0 t C *expi 0 t (t) Re D expi 0 t Do you reeber how the A, B, C, D constants are related? If not, go bac and review until it becoes second nature! 9

3 Position: x(t) Acos 0 t A, are unnown constants - ust be deterined fro initial conditions, in principle, is nown and is a characteristic of the physical syste Velocity: Acceleration: dx dt x(t) Asin t 0 0 d 2 x dt x(t) Acos 0 t 0 2 x(t) This type of pure sinusoidal otion with a single frequency is called SIMPLE HARMONIC MOTION 17

4 REVIEW MASS ON IDEAL SPRING F(x) x F(x) x Newton Particular type of force., nown x x x x x 0 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 paraeter that describes tie "dot" and "double dot" ean differentiate w.r.t. tie, are nown constants 13

5 REVIEW MASS ON IDEAL SPRING x x 0 x x(t) Ce pt C, p are unnown (for now) constants, possibly coplex x(t) p 2 Ce pt p 2 x(t) Substitute: p 2 x x 0 p i i 0 p is now nown. Note that 0 is NOT a new quantity! It is just a rewriting of old ones - partly shorthand, but also 14 eans frequency to physicists!

6 x(t) Acos 0 t A, chosen to fit initial conditions: x(0) = x 0 and v(0) = v 0 x 0 Acos x v 0 0 Asin Square and add: Divide: x 2 0 v A2 0 cos 2 sin 2 A 2 v 0 0 x 0 tan 15

7 x(t) Acos 0 t x(t) x 2 0 v 2 0 cos 2 0 t arctan v 0 0 x 0 x(t) Acos cos 0 t Asin sin 0 t x(t) Aei 2 ei 0t x(t) Re Ae i e i 0t Aei 2 ei 0t 2 arbitrary constants (A, because 2nd order linear differential equation 16

8 THE LC CIRCUIT L V L V C 0 Kirchoff s law (not Newton this tie) I C q V L L di dt L d 2 q dt 2 Lq V C q C q 1 LC q

9 Free, undaped oscillators other exaples L x r; r L No friction x x I C q q 1 LC q Coon notation for all T g L 02 0 g

10 THE DAMPED HARMONIC OSCILLATOR Reading: Main 3.1, 3.2, 3.3 Taylor 5.4 Giancoli 14.7, 14.8

11 Free, undaped oscillators other exaples L x r; r L No friction x x I C q q 1 LC q Coon notation for all T g L 02 0 g

12 friction x LI 1 C q RI 0 x x bx Lq 1 C q R q 0 r L c Coon notation for all T g L b' g

13 Natural otion of daped haronic oscillator Force ẋ restoring force resistive force x x Need a odel for this. Try restoring force x proportional to velocity bx How do we choose a odel? Physically reasonable, atheatically tractable Validation coes IF it describes the experiental syste accurately

14 Natural otion of daped haronic oscillator Force ẋ restoring force resistive force x x bx x Divide by coefficient of d 2 x/dt 2 and rearrange: x 2 x 0 2x 0 inverse tie and (rate or frequency) are generic to any oscillating syste This is the notation of TM; Main uses = 2.

15 Natural otion of daped haronic oscillator x 2 x 2 0 x 0 Try x(t) Ce pt C, p are unnown constants x (t) pxt, x (t) p 2 x(t) Substitute: p p 0 x(t) 0 Now p is nown (and there are 2 p values) p x(t) Ce p t C'e p t Must be sure to ae x real!

16 Natural otion of daped HO Can identify 3 cases 0 underdaped 0 overdaped 0 critically daped tie --->

17 underdaped tie ---> p i 1 x(t) Ce ti 1 t C * e ti 1 t Keep x(t) real x(t) Ae t cos 1 t coplex <-> ap/phase Syste oscillates at "frequency" 1 (very close to 0 ) - but in fact there is not only one single frequency associated with the otion as we will see.

18 underdaped 0 Daping tie or "1/e" tie is = 1/ (>> 1/ if is very sall) How any T 0 periods elapse in the daping tie? This nuber (ties π) is the Quality factor or Q of the syste. Q T large if is sall copared to 0

19 overdaped 0 p Two values of p - both real x(t) C 1 e t C 2 e t Exponential daping with long tie constant ->

20 critically daped; quicest way to get to equilibriu 0 p One real value of p x(t) C 1 C 2 e t Exponential daping =1/1/, with at ost one oscillation