Presentation is loading. Please wait.

Presentation is loading. Please wait.

1 421: Oscillations 2 PH421: Homework 30%; Laboratory reports 35%; Final 35%. All lab reports will be submitted in class. ~ November 2015 ~ MonTueWedThuFri.

Similar presentations


Presentation on theme: "1 421: Oscillations 2 PH421: Homework 30%; Laboratory reports 35%; Final 35%. All lab reports will be submitted in class. ~ November 2015 ~ MonTueWedThuFri."— Presentation transcript:

1

2 1 421: Oscillations

3 2 PH421: Homework 30%; Laboratory reports 35%; Final 35%. All lab reports will be submitted in class. ~ November 2015 ~ MonTueWedThuFri 91011 Civic Holiday 12 Lab & Discussion: the LCR circuit 13 Upload data -HW1 due 161718 -HW2 (1,2) due 1920 -HW2 due Formal LCR Lab Report Due 23 Upload data 24 Fourier Methods & Impulse Lab 2526 Civic Holiday 27 Civic Holiday 311 Research in the Physics; intro to senior thesis 2 -demo pendulum lab HW3(1,2) due -Formal Fourier Impulse Lab Due 3 4 Review Session I HW3 due

4 Goal #1: Intro to Formal Technical Writing Two “formal” lab reports (35%) are required. Good technical writing is very similar to writing an essay with sub-heading. We want to hear a convincing story, not a shopping list of everything you did. Check out course web-site 3

5 Goal #2: Intro to Research in Physics We introduce the senior thesis research/writing requirement If you’re thinking about about grad school, med school, etc. and have not started/planned out research opportunities you are already well behind the competition. Start now! -due Feb. Department SURE Science scholarship -due (very soon) external competitions, REUs, etc. 4

6 5 Are oscillations ubiquitous or are they merely a paradigm? Superposition of brain neuron activity

7 REPRESENTING SIMPLE HARMONIC MOTION http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/shm.gif 6 not simple simple

8 y(t) Simple Harmonic Motion Watch as time evolves 7

9 -A A amplitude phase angle determined by initial conditions period angular freq (cyclic) freq determined by physical system 8

10 Position (cm) Velocity (cm/s) Acceleration (cm/s 2 ) time (s) 9

11 These representations of the position of a simple harmonic oscillator as a function of time are all equivalent - there are 2 arbitrary constants in each. Note that A, , B p and B q are REAL; C and D are COMPLEX. x(t) is real-valued variable in all cases. Engrave these on your soul - and know how to derive the relationships among A &  ; B p & B q ; C; and D. 10 A: B: C: D:

12 m = 0.01 kg; k = 36 Nm -1. At t = 0, m is displaced 50mm to the right and is moving to the right at 1.7 ms -1. Express the motion in form A form B x m m k k Example: initial conditions 11

13 12

14 m = 0.01 kg; k = 36 Nm -1. At t = 0, m is displaced 50mm to the right and is moving to the right at 1.7 ms -1. Express the motion in form C form D x m m k k Using complex numbers: initial conditions. Same example as before, but now use the "C" and "D" forms 13

15 14

16 15 Clicker Questions

17 16 A particle executes simple harmonic motion. When the velocity of the particle is a maximum which one of the following gives the correct values of potential energy and acceleration of the particle. (a)potential energy is maximum and acceleration is maximum. (b)potential energy is maximum and acceleration is zero. (c)potential energy is minimum and acceleration is maximum. (d)potential energy is minimum and acceleration is zero.

18 17 A particle executes simple harmonic motion. When the velocity of the particle is a maximum which one of the following gives the correct values of potential energy and acceleration of the particle. (a)potential energy is maximum and acceleration is maximum. (b)potential energy is maximum and acceleration is zero. (c)potential energy is minimum and acceleration is maximum. (d)potential energy is minimum and acceleration is zero. Answer (d). When velocity is maximum displacement is zero so potential energy and acceleration are both zero.

19 18 A mass vibrates on the end of the spring. The mass is replaced with another mass and the frequency of oscillation doubles. The mass was changed by a factor of (a)1/4 (b) ½ (c) 2 (d) 4

20 19 A mass vibrates on the end of the spring. The mass is replaced with another mass and the frequency of oscillation doubles. The mass was changed by a factor of (a)1/4 (b) 1/2 (c) 2 (d) 4 Answer (a). Since the frequency has increased the mass must have decreased. Frequency is inversely proportional to the square root of mass, so to double frequency the mass must change by a factor of 1/4.

21 20 A mass vibrates on the end of the spring. The mass is replaced with another mass and the frequency of oscillation doubles. The maximum acceleration of the mass: (a) remains the same. (b) is halved. (c) is doubled. (d) is quadrupled.

22 21 A mass vibrates on the end of the spring. The mass is replaced with another mass and the frequency of oscillation doubles. The maximum acceleration of the mass: (a) remains the same. (b) is halved. (c) is doubled. (d) is quadrupled. Answer (d). Acceleration is proportional to frequency squared. If frequency is doubled than acceleration is quadrupled.

23 22 A particle oscillates on the end of a spring and its position as a function of time is shown below. At the moment when the mass is at the point P it has (a) positive velocity and positive acceleration (b) positive velocity and negative acceleration (c) negative velocity and negative acceleration (d) negative velocity and positive acceleration

24 23 A particle oscillates on the end of a spring and its position as a function of time is shown below. At the moment when the mass is at the point P it has (a) positive velocity and positive acceleration (b) positive velocity and negative acceleration (c) negative velocity and negative acceleration (d) negative velocity and positive acceleration Answer (b). The slope is positive so velocity is positive. Since the slope is getting smaller with time the acceleration is negative.

25 24 Optional Review of Complex Numbers

26 Real Imag a b  |z| Complex numbers Argand diagram 25

27 Euler’s relation 26

28 Consistency argument If these represent the same thing, then the assumed Euler relationship says: Equate real parts:Equate imaginary parts: 27

29 Real Imag t = 0, T 0, 2T 0 t = T 0 /4 t = t PHASOR 28

30 Adding complex numbers is easy in rectangular form Real Imag a b d c 29

31 Multiplication and division of complex numbers is easy in polar form Real Imag  |z| |w|   30

32 Real Imag a b Another important idea is the COMPLEX CONJUGATE of a complex number. To form the c.c., change i -> -i The product of a complex number and its complex conjugate is REAL. We say “zz* equals mod z squared” |z||z|  31

33 And finally, rationalizing complex numbers, or: what to do when there's an i in the denominator? 32


Download ppt "1 421: Oscillations 2 PH421: Homework 30%; Laboratory reports 35%; Final 35%. All lab reports will be submitted in class. ~ November 2015 ~ MonTueWedThuFri."

Similar presentations


Ads by Google