Page 1. Physics 131: Lecture 22. Today s Agenda. SHM and Circles. Position

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1 Physics 3: ecture Today s genda Siple haronic otion Deinition Period and requency Position, velocity, and acceleration Period o a ass on a spring Vertical spring Energy and siple haronic otion Energy o a spring orce Physics 0: ecture, Pg peg on a rotating turntable undergoes periodic otion There is a close relationship between circular otion and siple haronic otion SHM and Circles Physics 0: ecture, Pg ngular position: t Position Call the vector pointing toward the peg The x-position o is x cos cos t This is our equation or SHM! Physics 0: ecture, Pg 3 Page

2 Velocity The x-coponent o v is: v x v sin v r v sin t vax v t Physics 0: ecture, Pg 4 Position and Velocity t x cos v sin x is zero when v = + or - v is zero when x = + or - Physics 0: ecture, Pg 5 The x-coponent o a is: a x a cos a r a cos t a ax cceleration a t Physics 0: ecture, Pg 6 Page

3 Siple Haronic Motion: x(t) = []cos(t) x(t) = []sin(t) v(t) = -[]sin(t) v(t) = []cos(t) OR a(t) () = -[ ]cos(t) a(t) () = -[ ]sin(t) x ax = v ax = a ax = Period = T (seconds per cycle) Frequency = = /T (cycles per second) ngular requency = = = /T Physics 0: ecture, Pg 7 Siple Haronic Motion: x(t) = []cos(t) v(t) = -[]sin(t) a(t) () = -[ ]cos(t) x(t) = []sin(t) v(t) = []cos(t) a(t) = -[ ]sin(t) v t = 0 x= x ax = v= 0 t = 0 x= 0 v= v ax = Physics 0: ecture, Pg 8 Clicer Question : n 0.80 g object is attached to one end o a spring, and the syste is set into siple haronic otion. The displaceent o x o the object as a unction o tie is shown in the drawing. What is the correct equation or x? (a) x cos t (b) (c) (d) x 0.60 sin t x sin t x 0.60 sin t Physics 0: ecture, Pg 9 Page 3

4 Clicer Question : Object is attached to spring and is oving in siple haronic otion. Object B is attached to spring B and is oving in siple haronic otion. The period and the aplitude o object B are both two ties the corresponding values or object. How do the axiu speeds o the two objects copare? a) The axiu speed o is one ourth that o object B. b) The axiu speed o is one hal that o object B. c) The axiu speed o is the sae as that o object B. d) The axiu speed o is two ties that o object B. e) The axiu speed o is our ties that o object B. Physics 0: ecture, Pg 0 Phase Constant Which one? xt cost x0 cos 0 cos vt sin t v 0 sin v0 sin 3 x t cost 3 Physics 0: ecture, Pg Period or a Mass on a Spring What can Newton s second law tell us about SHM? F a x a cost cost ω T Physics 0: ecture, Pg Page 4

5 Clicer Question.5: ass is attached to a spring. I pull it distance o and it oscillates with requency. I I pull it a distance o what will the requency be? (a) 4 (b) (c) (d) / (e) /4 Physics 0: ecture, Pg 3 Clicer Question 3: Playing a tune on wine glasses is an application o SHM. To change the requency o sound you need to adjust the aount o water. I you wanted to increase the requency, what would you do? a) dd ore water to the glass. b) Reove soe water ro the glass. Physics 0: ecture, Pg 4 Clicer Question 4: bloc o ass oscillates on a horizontal spring with period T.0 s. I a second identical bloc is glued to the top o the irst bloc, the new period will be. 0.0 s. B..4 s. C..0 s. D..8 s. E. 4.0 s. Physics 0: ecture, Pg 5 Page 5

6 Clicer Question 4: bloc o ass oscillates on a horizontal spring with period T.0 s. I a second identical bloc is glued to the top o the irst bloc, the new period will be Physics 0: ecture, Pg 6 Clicer Question 5: Two identical blocs oscillate on dierent horizontal springs. Which spring has the larger spring constant?. The red spring. B. The blue spring. C. There s not enough inoration to tell. Physics 0: ecture, Pg 7 Clicer Question 6: ass on a spring oscillates bac & orth with siple haronic otion o aplitude. plot o displaceent (x) versus tie (t) is shown below. t what points during its oscillation is the total echanical energy (K+U) o the ass and spring a axiu? (Ignore riction). a) When x = + or - (i.e. axiu displaceent) b) When x = 0 (i.e. zero displaceent) c) The echanical energy o the syste is constant x + - Physics 0: ecture, Pg 8 t Page 6

7 Energy Conservation I there are no non-conservative orces acting, the echanical energy will be conserved: E K U v x t axiu displaceent, x=, v = 0: E t zero displaceent, x = 0: ax E v E ω Physics 0: ecture, Pg 9 Energy Conservation This ay also be shown ore orally: E v x Find this as a unction o tie: sin t t E cos t t t t t t E sin cos E sin cos E sin cos Physics 0: ecture, Pg 0 Clicer Question 7: ass oscillates in siple haronic otion with aplitude. I the ass is doubled, but the aplitude is not changed, what will happen to the total energy o the syste? a) total energy will increase b) total energy will not change c) total energy will decrease Physics 0: ecture, Pg Page 7

8 K Exaple bloc o ass 5.0 g on a rictionless surace is attached to a horizontal spring with = 3 N/. I the bloc is pulled 0.0 ro the equilibriu position o the spring, what velocity will it have when it is 0.0 ro the equilibriu position? v K U K U i i vi xi v K i U U i x x i 0.44/s x 0.64J 5g0.44/s 0.64J 3N/ 0.0 3N/ 0.0 Physics 0: ecture, Pg Pendulu Motion Free-body diagra or pendulu: g ω g y T g x g sin T g cos Period and requency do not depend on, or! (For sall oscillations) g Physics 0: ecture, Pg 3 Clicer Question 8: pendulu is hanging vertically ro the ceiling o an elevator. Initially the elevator is at rest and the period o the pendulu is T. Now the pendulu accelerates upward. The period o the pendulu will now be. greater than T B. equal to T C. less than T g T g Physics 0: ecture, Pg 4 Page 8

Clicer Question 9: ass is suspended ro the ceiling o an elevator by a spring. When the elevator is at rest, the period is T. What happens to the period when the elevator is accelerating upward?

9 Clicer Question 9: ass is suspended ro the ceiling o an elevator by a spring. When the elevator is at rest, the period is T. What happens to the period when the elevator is accelerating upward? a) period will increase b) period will not change c) period will decrease Physics 0: ecture, Pg 5 Vertical Mass and Spring I we include gravity, there are two orces acting on ass. With ass, new equilibriu position has spring stretched d. F y = 0 d g = 0 d = g/ et this point be y = 0 F F = a (d - y) g = a (g/) - y g = a - y = a Sae as horizontal! SHO New equilibriu position y = -d Physics 0: ecture, Pg 6 Proble g bloc hangs ro a spring with spring constant 000 N/. The bloc is pulled down 5.0 c ro the equilibriu position and given an initial velocity o.0 /s bac towards equilibriu. What are the (a) requency, (b) aplitude and (c) total echanical energy o the otion? Physics 0: ecture, Pg 7 Page 9

Page 1. Physics 131: Lecture 22. SHM and Circles. Today s Agenda. Position. Velocity. Position and Velocity. Acceleration. v Asin.

Page 1. Physics 131: Lecture 22. SHM and Circles. Today s Agenda. Position. Velocity. Position and Velocity. Acceleration. v Asin. Physics 3: ecture Today s enda Siple haronic otion Deinition Period and requency Position, velocity, and acceleration Period o a ass on a sprin Vertical sprin Enery and siple haronic otion Enery o a sprin