Unit 14 Harmonic Motion. Your Comments

Transcription

1 Today s Concepts: Periodic Motion Siple - Mass on spring Daped Forced Resonance Siple - Pendulu Unit 1, Slide 1 Your Coents Please go through the three equations for siple haronic otion and phase angle The concept of relating siple haronic otion and unifor circular otion was very confusng i thin i understand ost of it but would lie to go through soe eaples Please go over the forulas and how they relate to pi/, pi/4 and so on. Also please go over prelecture question. I a not sure how you got that answer. I need help with the phase angles and interpreting the graphs for haronic otion. Please stop sipping steps during in class probles. Please go through each step, even if it sees silly to you. It's easier to get a visual by you writing it out, it gets confusing when I don't understand how you derived soething. Please go over frequency of oscillations can we review question fro the prelecture? Also question three in the bridge question entitled, "Aplitude and period in siple haronic otion" Unit 1, Slide 1

2 Periodic Motion Any otion that repeats itself over and over again. Period, T: tie it taes to coplete one cycle Frequency, f: nuber of cycles copleted every second f = 1/T Unit 1, Slide 3 Siple -A 0 A = A*cos(wt +f) Unit 1, Slide 4

3 Yesterday s Quiz A cart of ass = 3.5 g carrying a spring and oving at speed v = 3.6 /s hits a stationary cart of ass M = 10 g. Assue all otion is along a line and the collision is totally elastic a) Before the collision, what is the speed of the center of ass of this syste? b) Before th collision, what is the echanical energy of the syste? c) After the collision is over, what is the velocity of? d) After the collision is over, what is the velocity of M? e) After the collision is over, what is the speed of the center of ass of this syste? f) After the collision, what is the echanical energy of the syste? - Slide 5 You had this question A 11 long string is wrapped around a pulley that is free to rotate about a fied ale. The pulley has a radius of 5.8 c and is initially at rest. The thicness and ass of the string are negligible. Soeone pulls on the free end of the string with a 0N force, winding it with a constant acceleration of 16 rad/sec. 1) Through what angle has the pulley rotated when the string is copletely unwound? ) What is final angular velocity of the pulley - Slide 6 3

4 Lie shadow of rotating ball v tan = Aw v tan- = -Aw*(-sin(Q)) = -Aw*sin(wt) a cen = Aw a cen- = Aw *(-cos(q)) = -Aw *cos(wt) - Slide 7 What is w? Tells you how fast the object oscillates. wt = p w = p/t w = pf What is A? Aplitude aiu stretch of the spring. Unit 1, Slide 8 4

5 What do they ean graphically? T -p -p A p p Change A Unit 1, Slide 9 What do they ean graphically? T o -p -p A p p T 1 Change w and T Unit 1, Slide 10 5

6 What do they ean graphically? T -p -p A p p Change f Unit 1, Slide 11 Question A sall rubber ball is dropped fro a height of 10 eters. It hits the floor and rebounds to the height of 10 eters, over and over again. This is: A. Periodic otion but not siple haronic otion B. Siple haronic otion but not periodic otion C. Both siple haronic and periodic otion D. Neither siple haronic or periodic otion Unit 1, Slide 1 6

7 Eaple 14.1 (ass on spring) -A 0 A A g ass is attached to a spring with a spring constant of =5N/. The spring is stretched 0c and then released at t=0. a) What is the period of the ass? b) What is the equation of otion of the ass? Unit 1, Slide 13 Eaple 14. (ass on spring - velocity) -A 0 A A g ass is attached to a spring with a spring constant of =5N/. The spring is stretched 0c and then released at t=0. a) What is it s aiu velocity? b) What is it s aiu acceleration? Unit 1, Slide 14 7

8 ChecPoint -A 0 A A ass on a spring oves with siple haronic otion as shown. Where is the acceleration of the ass ost positive? A) = -A B) = 0 C) = A Unit 1, Slide 15 Bridge Question A ass oscillates bac and forth on a spring. Its position as a function of tie is shown below. At which of the points shown does the ass have positive velocity and negative acceleration? (A) R (B) S (C) T (D) U (E) V Mechanics Lecture 1, Slide 16 8

9 Energy -A 0 A For ass on a spring we now PE = ½ For any object we now KE = ½ v What does that tell us in the case of a SHO? Mechanics Lecture 1, Slide 17 Eaple 14.3 (ass on spring - energy) -A 0 A A g ass is attached to a spring with a spring constant of =5N/. The spring is stretched 0c and then released at t=0. a) What is its total echanical energy? b) What is its aiu inetic energy? Mechanics Lecture 1, Slide 18 9

10 Question (A) (B) (C) A g ass is attached to a spring and oscillates bac and forth between the two ends of the arrow shown above. At what position is the inetic energy aiu? A) (A) B) (B) C) (C) D) the inetic energy is sae at all three points. Mechanics Lecture 1, Slide 19 Question (A) (B) (C) A g ass is attached to a spring and oscillates bac and forth between the two ends of the arrow shown above. At what position is the echanical energy aiu? A) (A) B) (B) C) (C) D) the echanical energy is sae at all three points. Mechanics Lecture 1, Slide 0 10

11 Eaple 14.5 (Raning) Consider four different systes, each ade of a bloc attached to an ideal horizontal spring. Ran the in order of their total echanical energy, fro largest to sallest value. (a) Bloc ass 0.50 g and spring constant 500 N/, with aplitude (b) Bloc ass 0.60 g and spring constant 300 N/, with speed 1.0 /s when passing through equilibriu. (c) Bloc ass 1. g and spring constant 400 N/, with speed 0.50 /s when passing through = (d) Bloc ass.0 g and spring constant 00 N/, with speed 0.0 /s when passing through =0.050., Slide 1 Daped Physical oscillator coe to rest due daping force F D = -bv SF = a - -b d/dt = d /dt Solution: (t) = a e -bt/ cos(w t +f) ω = b 4 Mechanics Lecture 1, Slide 11

12 Daped Notice we re taing the square root of a difference. What if it s negative? ω = b 4 / > (b/) / < (b/) / = (b/) under daped over daped critically daped (w = 0) no oscillations Mechanics Lecture 1, Slide 3 Daped Overdaped Critically daped Under daped Which one do you want for your shocs on your car? Mechanics Lecture 1, Slide 4 1

13 Underdaped (t) = a e -bt/ cos(w t +f) Lie a SHO with a tie dependent aplitude Mechanics Lecture 1, Slide 5 Forced w d Add third force restoring force (e.g. spring) - daping force (e.g. air drag) -bv driving force (e.g. otor) F a cos(w d + f) Aplitude of ass = F a ( (w o - w d ) + b w d )) 0.5 Natural Frequency of ass (w = /) Frequency of driving force Mechanics Lecture 1, Slide 6 13

14 Question w d Aplitude of ass F a ( (w o - w d ) + b w d )) 0.5 At what frequency should the driving force oscillate in order for the aiu aplitude for the ass? A. w d = 0/sec B. w d = w o = C. w d = infinity Resonance! Mechanics Lecture 1, Slide 7 = I Pendulu R CM For sall - MgX CM - MgR CM d MgR = - dt I d dt w = d I dt CM = -w MgR CM I X CM arc-length = R CM R CM X CM Mg Mechanics Lecture 8, Slide 8 14

15 L Siple Pendulu = A*cos(wt +f) Q = Q a *cos(wt +f) = I g*(l*sinq)=l * Mechanics Lecture 8, Slide 9 Eaple 14.8 (Siple Pendulu) L A 00gra ass is attached to the end of eter long string and allowed to swing bac and forth. What is it s period, T? Mechanics Lecture 8, Slide 30 15