Classical Mechanics Lecture 22

Transcription

1 Classical Mechanics Lecture 22 Today s Concept: Siple Haronic Mo7on: Mo#on of a Pendulu Mechanics Lecture 8, Slide 1

2 Grading Unit 14 and 15 Ac7vity Guides will not be graded Please turn in:! Unit 14 WriIen Hoework on Mon, Dec 5! The Mini-labbook on your SHM or Karate Project, Dec 12 The last FlipItPhysics lecture is a bonus.! not on exa! use it if you issed any points earlier! Also use the extra Tipler & Mosca ques7ons on F.P. for bonus and prep Final Exa 7e: Tue., Dec. 12, 3:30 p Final exa roo: 2600

3 Your Coents How do we know when to use the sin forula or cos forula for the posi7on of the oscilla7ng syste? Are they both viable? also can you see this face --> ( ʖ )? this is kind of fro the last lecture but how do you get the equa7on of x= A sin(ωt+φ) fro a word proble? (ie. How do you know waht φ is? and is it cos or sin? ) Who's the genius who decided oega should have two eanings? Did they run out of Greek leiers? Why don't they fly over there and get soe ore? It would probably help boost their econoy at this point. I a finding it difficult to understand how the oent of iner7a and the radius are both being used in the equa7on. Isn't the oent of iner7a dependent on the radius? despacito talking about haronic o7ons, Lets all dance "GANGNOM style ", its a perfect prac7cal exaple! For the torsion pendulu, what did the lower case kappa (κ, κ) represent? What causes that constant? Thanks. How do you know when to use what forula? In the prelecture they didn't explain clearly if you can use the sae forula for a pendulu with a ass aiached to the end as for a pendulu without a ass aiached to it. Mechanics Lecture 8, Slide 2

4 There is a theory which states that if ever anybody discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by soething even ore bizarre and inexplicable. Text There is another theory which states that this has already happened. Douglas Adas Mechanics Lecture 8, Slide 3

5 I want to know why the answer to life is 42! Drill a hole through the earth and jup in what happens? Just for fun you don t need to know this.

6 I want to know why the answer to life is 42! Drill a hole through the earth and jup in what happens? You will oscillate like a ass on a spring with a period of 84 inutes. It takes 42 inutes to coe out the other side! k = g/r E Mechanics Lecture 8, Slide 5

7 I want to know why the answer to life is 42! Drill a hole through the earth and jup in what happens? You will oscillate like a ass on a spring with a period of 84 inutes. It takes 42 inutes to coe out the other side! The hole doesn t even have to go through the iddle you get the sae answer anyway as long as there is no fric7on. Mechanics Lecture 8, Slide 6

8 I want to know why the answer to life is 42! This is also the sae period of an object orbi7ng the earth right at ground level. Just for fun you don t need to know this. Mechanics Lecture 8, Slide 7

9 Panic! Is there such a thing as Rota7onal Haronic Mo7on? There beier not be... Yes there is. Are you ready?

10 Torsion Pendulu wire τ θ I Q: In the prelecture the equa7on for restoring torque is given as τ=-κθ in clockwise direc7on..so if the restoring torque is in counter clockwise direc7ons then would τ be posi7ve? Mechanics Lecture 8, Slide 8

11 CheckPoint A torsion pendulu is used as the 7ing eleent in a clock as shown. The speed of the clock is adjusted by changing the distance of two sall disks fro the rota7on axis of the pendulu. If we adjust the disks so that they are closer to the rota7on axis, the clock runs: A) Faster B) Slower Sall disks Mechanics Lecture 8, Slide 9

12 CheckPoint If we adjust the disks so that they are closer to the rota7on axis, the clock runs A) Faster B) Slower A) The oent of iner7a decreases, so the angular frequency increases, which akes the period shorter and thus the clock faster. B) T = 2pi * sqrt(i/mgrc). If Rc decreases, T will increase, aking the clock run slower. Mechanics Lecture 8, Slide 10

13 Pendulu θ R CM For sall θ X CM Mg θ R CM X CM arc-length = R CM θ Mechanics Lecture 8, Slide 11

14 pivot The Siple Pendulu θ R CM CM θ L The siple case The general case Mechanics Lecture 8, Slide 12

15 CheckPoint If the clock is running too fast, the weight needs to be oved A) Up B) Down If the clock is running too fast then we want to reduce it's period, T, and to do that we need to increase oega, the frequency it oves with and to do that we need the posi7on of the center of ass to be further fro the pivot, which is achieved by oving the weight down. Mechanics Lecture 8, Slide 14

16 The Stick Pendulu pivot θ R CM CM M Sae period Mechanics Lecture 8, Slide 15

17 Case 1 Case 2 CheckPoint In Case 1 a s7ck of ass and length L is pivoted at one end and used as a pendulu. In Case 2 a point par7cle of ass is aiached to the center of the sae s7ck. In which case is the period of the pendulu the longest? A) Case 1 B) Case 2 C) Sae C is not the right answer. Lets work through it Mechanics Lecture 8, Slide 16

18 Case 1 Case 2 In Case 1 a s7ck of ass and length L is pivoted at one end and used as a pendulu. In Case 2 a point par7cle of ass is aiached to a string of length L/2? In which case is the period of the pendulu longest? A) Case 1 B) Case 2 C) Sae T = 2 s 2 3 L g T = 2 s 1 2 L g Mechanics Lecture 8, Slide 17

19 T 2 Suppose you start with 2 different pendula, one having period T 1 and the other having period T 2. T 1 T 1 > T 2 Now suppose you ake a new pendulu by hanging the first two fro the sae pivot and gluing the together. What is the period of the new pendulu? A) T 1 B) T 2 C) In between Mechanics Lecture 8, Slide 18

20 Case 1 Case 2 In Case 1 a s7ck of ass and length L is pivoted at one end and used as a pendulu. In Case 2 a point par7cle of ass is aiached to the center of the sae s7ck. In which case is the period of the pendulu the longest? A) Case 1 B) Case 2 C) Sae Now lets work through it in detail Mechanics Lecture 8, Slide 19

21 Case 1 Case 2 Lets copare for each case. Mechanics Lecture 8, Slide 20

22 Case 1 Case 2 Lets copare for each case. (A) (B) (C) Mechanics Lecture 8, Slide 21

23 So we can work out Case 1 Case 2 In which case is the period longest? A) Case 1 B) Case 2 C) They are the sae Mechanics Lecture 8, Slide 22

24 The Sall Angle Approxiation θ R CM - Exact expression X CM arc-length = R CM θ % difference between θ and sinθ Angle (degrees) Mechanics Lecture 8, Slide 23

25 Clicker Question A pendulu is ade by hanging a thin hoola-hoop of diaeter D on a sall nail. What is the angular frequency of oscilla7on of the hoop for sall displaceents? (I CM = R 2 for a hoop) A) pivot (nail) B) D C) Mechanics Lecture 8, Slide 24

26 The angular frequency of oscilla7on of the hoop for sall displaceents will be given by Use parallel axis theore: I = I CM + R 2 = R 2 + R 2 = 2R 2 pivot (nail) R So X CM Mechanics Lecture 8, Slide 25