Classical Mechanics Lecture 24

Transcription

1 Classical Mechanics Lecture 24 Today s Concepts: A) Superposi6on B) Standing Waves Mechanics Lecture 24, Slide 1

2 Case A y CheckPoint v x y Case B v x Suppose a pulse in Case A described by the func6on y(x,t) = P(x vt). Which of the following func6ons described the pulse in Case B? A) y(x,t) = P(x + vt) B) y(x,t) = P(x + vt) C) y(x,t) = P(x vt) Mechanics Lecture 24, Slide 4

3 y A) y(x,t) = P(x + vt) B) y(x,t) = P(x + vt) C) y(x,t) = P(x vt) Case A Case B y v v x x A) Everything is the same except the direc6on of the velocity is in the opposite direc6on. B) x+vt makes the wave go in the nega6ve x direc6on, - P flips the wave upside down C) This is the only equa6on that the waves will move to the leq. Mechanics Lecture 24, Slide 5

4 Superposition Q: What happens when two waves collide? A: They ADD together! Movie (super_pulse) Movie (super_pulse2) Mechanics Lecture 24, Slide 6

5 Superposition The wave equa6on we derived last 6me is linear. (It has no terms where the variables x,t are squared.) For linear equa6ons, if we have two separate solu6ons, A and B, then A + B is also a solu6on! A(ω 1 t) B(ω 2 ω 2 > ω 1 C(t) = A(t) + CONSTRUCTIVE INTERFERENCE DESTRUCTIVE INTERFERENCE Mechanics Lecture 24, Slide 7

6 Beats Can we predict this pacern mathema6cally? Of course! Just add two cosines and remember the iden6ty: where and cos(ω L t) cos(ω H t) hcp:// Mechanics Lecture 24, Slide 8

7 Standing Waves What happens when two waves having the same frequency but moving in the opposite direc6on meet? Movie (super) Wave 1 Wave 2 Superposi6on zero (Nodes) Wave 2 Wave 1 Wave 2 Wave 1 Mechanics Lecture 24, Slide 9

8 How it Works How to make it: Sta6onary wave Changing amplitude Demo Mechanics Lecture 24, Slide 10

9 CheckPoint Suppose the strings on your guitar are 24 long as shown. The frets are the places along the neck where you can put your finger to make the wavelength shorter, and appear as horizontal white lines on the picture. When no frets are being pushed the frequency of the highest string is 4 6mes higher than the frequency of the lowest string. Is it possible to play the lowest string with your finger on any of the frets shown and hear the same frequency as the highest string? A) Yes B) No frets Mechanics Lecture 24, Slide 11

10 Clicker Question If you want to increase the frequency of a 24 string by a factor of 4 while keeping the tension the same, how long should the string be? A) 4 B) 6 C) 8 D) 12 frets Mechanics Lecture 24, Slide 12

11 Clicker Question Is there a fret that you can push on this guitar to make the string 6 long? A) Yes B) No frets Mechanics Lecture 24, Slide 13

12 Clicker Question The highest string vibrates with a frequency that is 4 6mes that of the lowest string. Compare the speed of a wave on the high string and the low string: A) v high = v low* 2 B) v high = v low* 4 C) v high = v low* 16 v low v high Mechanics Lecture 24, Slide 14

13 Clicker Question The speed of a wave on the high string is 4 6mes the speed on the lowest string. If the tensions are the same, how does the mass per unit length of the strings compare: A) µ low = µ high* 2 B) µ low = µ high* 4 Recall C) µ low = µ high* 16 v low v high Mechanics Lecture 24, Slide 15

14 Clicker Question Two cylinders have the same length and are made from the same material. If the mass of the bigger one is 16 6mes the mass of the smaller one, how do their diameters compare? A) d low = d high* 2 L B) d low = d high* 4 C) d low = d high* 16 d big d small Mechanics Lecture 24, Slide 16

15 CheckPoint The 6 strings on a guitar all have about the same length and are stretched with about the same tension. The highest string vibrates with a frequency that is 4 6mes that of the lowest string. If the strings are made of the same material, how would you expect the diameters of the lowest and highest strings to compare? A) d low = d high* 2 B) d low = d high* 4 C) d low = d high* 16 d low d high Mechanics Lecture 24, Slide 17

16 The highest string vibrates with a frequency that is 4 6mes that of the lowest string. If the strings are made of the same material, how would you expect the diameters of the lowest and highest strings to compare? A) d low = d high* 2 B) d low = d high* 4 C) d low = d high* 16 B) Because the frequency is inversely propor6onal to the square root of the mass density, the mass density would be 16 6mes greater for the low string. So, the radius would be 4 6mes greater. C) Because of the square root sign in the denominator, if diameter is 16 6mes, mass is 16 6mes and speed is 1/4 6mes. So frequency is 1/4 6mes as wavelength is constant. d low d high Mechanics Lecture 24, Slide 18

17 We just found that the frequency of a string scales with its diameter if the tension is the same. This is really what you find if you examine a guitar:.050 d low d high.012 Mechanics Lecture 24, Slide 19

18 Different Example 12 String Guitar The lighter string in these pairs is one octave higher (2x frequency). Their diameters are about half of the adjacent strings Mechanics Lecture 24, Slide 20

19 Mechanics Lecture 24, Slide 21

20 Homework Problem v = L/t m = m/l Mechanics Lecture 24, Slide 22

21 Homework Problem Average speed = distance / 6me = 4A/P = 4Af Mechanics Lecture 24, Slide 23

22 Homework Problem Same thing again Mechanics Lecture 24, Slide 24

23 So long, and thanks for all the fish.