1. PHY 231 1 PHYSICS 231 Lecture 38: Resonances, beats and review Remco Zegers Question hours: Thursday 12:00-13:00 & 17:15-18:15 Helproom

2. PHY 231 2 standing waves in a rope both ends fixed n =2L/n or L=n n /2 F: tension in rope  : mass per unit length f 1 : fundamental frequency

3. PHY 231 3 Both ends open

4. PHY 231 4 One end open, one end closed even harmonics are missing!!!

5. PHY 231 5 example A simple flute is played by blowing air in on one side and the other end is open. The length of the tube can be varied manually (like a trombone). What are the frequencies of the first two possible harmonics if L=0.5m? If the length is made half of the original length, how will these change v=343m/s?

6. PHY 231 6 example A simple flute is played by blowing air in on one side and the other end is closed. The length of the tube can be varied manually (like a trombone). What are the frequencies of the first two possible harmonics if L=0.5m? If the length is made half of the original length, how will these change v=343m/s?

7. PHY 231 7 harmonics Generally speaking, many harmonics with different intensities can be present at the same time. L +

8. PHY 231 8 beats Superposition of 2 waves with slightly different frequency The amplitude changes as a function of time, so the intensity of sound changes as a function of time. The beat frequency (number of intensity maxima/minima per second): f beat =|f a -f b | DEMO

9. PHY 231 9 example Someone is trying to tune a guitar. One of the strings is supposed to have a frequency of 500 Hz. The person is using a tuning fork which produces a sound of exactly this frequency, but while sounding the fork and the playing the guitar, hears a beat in the sound with a frequency of 3 Hz (3 beat per second). a) What is the real frequency of the guitar string? b) By what fraction does the person need to change the tension of the guitar string to tune it properly?

10. PHY 231 10 Resonances Realistically, oscillations are damped due to frictional forces. However, we can drive the oscillation via an external source. Example: mass on a spring: natural frequency f=1/(2  )  (k/m) If the frequency of the driving force equals the natural frequency: large oscillations occur: Resonance demo Resonances occur in many daily situations: shock absorber in car playing basketball resonating lecture room!! Famous example: Tacoma bridge

11. PHY 231 11 see also recitations on Thursday (12-1 and 17:15-18:15) review on Friday any more??? today: one problem each from ch. 2,3,4,5,6 Friday: one problem each from 7,8,9 and the rest 10,11,12,13,14 (2 each) Start with the sample problems on the web to see how you stand on each chapter!!

12. PHY 231 12 chapter 2. A person throws 2 stones from the top of a building with a speed of 20 m/s. One is thrown up, and the other is thrown down. The first one hits the street after 5 s. How much later does the second one hit? x(t)=x(0)+v(0)t+½at 2 v(t)=v(0)+at

13. PHY 231 13 chapter 3 v0v0 30m 3m A car is trying to jump over a 30m-wide river using a ramp of 3 m high set at an angle of 30 0 with the horizontal. a) What is the minimum velocity v 0 required to cross the bridge? b) What is the highest point of the car?  =30 0

14. PHY 231 14 chapter 4 A person is pushing an ice-sledge of 50kg over a frozen lake with a force of 100N to the east. A strong wind is pushing from the south-west and produces a force of20N on the sledge. a) What is the acceleration of the sledge? b) if the coefficient of kinetic friction is 0.05, what is the acceleration? make sure you understand

15. PHY 231 15 chapter 5 A crate of 50kg is starting to slide from a slope. When it reaches the bottom, it is caught by a spring with a spring constant of 1000 N/m. a) If the crate was originally at a height of 10 m and friction can be ignored, how much is the spring maximally compressed? b) if the frictional force is 100N and the length of the slope is 15m, what is the maximal compression?

16. PHY 231 16 chapter 6 Two objects collide head on. Object 1 (m=5kg) has an initial velocity of 10m/s and object 2 (m=10 kg) has an initial velocity of -8 m/s. What is the resulting velocity a)of the combined object if the collision is fully inelastic? b)of each of the objects if the collision if fully elastic?