1. Vibrations and Waves by Peyton Garman and Caroline Littel

2. Authors’ Biography Peyton is a 18 year-old high school student from the Ghost Town of WhereTheHeck, Texas. Her sister, Ambrielle, and her share a common favorite color: Blue. She loves books and hates tequila. Caroline Littel is a 17 year-old high school student from Pleasant What, Tennessee. Her favorite color is purple, her least favorite dog breed is chihuahua (even though she technically believes them to be cats), and hates spiders.

3. Hooke’s Law Periodic motion is any action repeated in equal periods of time. Examples of this include a child on a playground swing, a pendulum, or a spring. At the equilibrium, speed reaches a maximum. After a spring is stretched and released, when it passes its equilibrium position, the speed reaches its highest point. At maximum displacement, spring force and acceleration reach a maximum. As the spring travels to the point furthest from its point of equilibrum.

4. Hooke’s Law In simple harmonic motion, restoring force is proportional to displacement. In simple harmonic movement, a restoring force is the equal and opposite reaction for the action that just took place. This is called Hooke’s Law and its formula is as follows: F elastic = -kx In this instance, spring force = (-spring constant)(displacement) Simple harmonic movement is essentially any periodic motion that is the result of a restoring force proportional to displacement. A stretched or compressed spring has elastic potential energy.

5. Practice Problem If a mass of 0.55 kg attached to a vertical spring stretches the spring 2.0 cm from its original equilibrium position, what is the spring constant?

6. Practice Problem Step One: Define Analyse your problem and identify variables Given:m = 0.55 kgg = 9.81 m/s 2 x = -2.0 cm or -0.02 m Unknown:k = ? Diagram: F elastic F g

7. Practice Problem Step Two: Plan Choose an equation or situation: When the mass is attached to the spring, the equilibrium position changes. At the new equilibrium position, the net force acting on the mass is zero. So, the spring force (given by Hooke’s law) must be equal and opposite to the weight of the mass. F net = 0 = F elastic + F g F elastic = -kx F g = -mg -kx - mg = 0 Rearrange the equation to isolate the unknown: kx = -mg k = -mg/x

8. Practice Problem Step Three: Calculate Substitute the values into the equation and solve: k = k = 270 N/m -(.55 kg)(9.81 m/s 2 ) -0.020 m

9. Practice Problem Step Four: Evaluate The value of k implies that 270 N of force is required to displace the spring one meter.

10. The Simple Pendulum A pendulum consists of a mass called a bob which is attached to a fixed string. When studying a simple pendulum, air resistance and friction are disregarded. When studying a physical pendulum, however, all possible factors must be taken into account. The restoring force of a pendulum is a component of the bob’s weight.

11. Amplitude, Frequency, and Period The maximum displacement from the equilibrium position is called amplitude. The period, T, is the time it takes for a pendulum’s complete cycle of motion to occur. If one complete cycle takes 20 s, the period is 20 s The number of complete cycles the pendulum swings is called the frequency. If one complete cycle is 20s, then the frequency is 1/20 cycles/s or 0.05 cycles/s. The SI unit for frequency is hertz. f = = = 0.05 Hz 1 T 20s

12. Practice Problems 1. If it takes one minute and thirty-seven seconds for a trapeze artist to finish a cycle, what is his or her frequency? a. 1/97 Hz or 0.01 Hz 2. If the frequency of a pirate ship ride at Summerfest is 0.035 Hz, how long does it take the ride to complete a cycle? If the duration of the ride is a total of 25 cycles, how long does it take to ride the ride from start to finish in minutes? a. One cycle takes 28.57 seconds b. The entire ride (25 cycles) takes 714.29 seconds or 11.90 minutes or 8 minutes and 54 seconds

13. Amplitude, Frequency, and Period The period of a simple pendulum depends on the pendulum length and free-fall acceleration. Although both a simple pendulum and a mass-spring system vibrate with simple harmonic motion, calculating the period and frequency of each requires a separate equation. Period of a simple pendulum

14. Terms to Know Medium- a physical environment through which a disturbance can travel Mechanical waves- wave that requires a medium through which to travel Transverse wave- a wave whose particles vibrate perpendicularly to the direction the wave is traveling Crest- the highest point above the equilibrium position. Trough- the lowest point below the equilibrium position. Wavelength or λ (lambda)- distance between 2 adjacent similar points of a wave, such as from crest to crest or from trough to trough

15. Terms to Know Pulse wave- a wave that consists of a single traveling pulse. Periodic wave- more than one pulse wave put together. Sine wave- a wave whose source vibrates with simple harmonic motion waveform- a curve showing the shape of a wave at a given time. Longitudinal wave- wave whose particles vibrate parallel to the direction the wave is traveling.

16. Labeling a Wavelength Medium Height

17. The Difference This is a pulse wave kkk This is a periodic wave

18. Waveform The waveform is the shape of a wave as it travels. There are many different waveforms. They are usually a shape which is repeated over and over (a "periodic waveform"). A common waveform is the sine wave. It is normally not possible to see a waveform without some device.shapewavesine wavedevice

19. Sine Wave A sine wave is a curve with this shape:curveshape This is a picture of a sine wave. All waves can be made by adding up sine waves.wavesadding

20. Transverse Wave In a transverse wave the particle displacement is perpendicular to the direction of wave propagation. The animation below shows a one-dimensional transverse plane wave propagating from left to right. The particles do not move along with the wave; they simply oscillate up and down about their individual equilibrium positions as the wave passes by. Pick a single particle and watch its motion.

21. Longitudinal Wave In a longitudinal wave the particle displacement is parallel to the direction of wave propagation. The animation shows a one-dimensional longitudinal plane wave propagating down a tube. The particles do not move down the tube with the wave; they simply move back and forth about their individual equilibrium positions. The wave is seen as the motion of the compressed region (ie, it is a pressure wave), which moves from left to right.

22. Difference in Transverse and Longitudinal Waves

23. Measuring Waves A wave can be measured in terms of its displacement from equilibrium. Height Displacement Equilibrium or medium

24. Measuring Wavelength The wavelength of a wave can be measured from one crest to the next crest or from one trough to the next trough. The amplitude of a wave can be measured from the origin to the crest or from the origin to the trough.

25. Example Problem a) wavelength _______cm b) amplitude ______ cm

26. Period, Frequency, and Wave Speed ➢ Period of a wave is the time for a particle on a medium to make one complete vibrational cycle. Period, being a time, is measured in units of time such as seconds, hours, days or years. ➢ Wave frequency describes the number of waves that pass a given point in a unit of time. ➢ Wave speed is an object refers to how fast an object is moving and is usually expressed as the distance traveled per time of travel. In the case of a wave, the speed is the distance traveled by a given point on the wave (such as a crest) in a given interval of time.

27. Formulas YOU WILL NEED! T= vibration λ= wavelength F= frequency (Units- Hz) V=speed of a wave

28. Example Problem Sachi is rock'n to her favorite radio station - 102.3 FM. The station broadcasts radio signals with a frequency of 1.023 x 108 Hz. The radio wave signal travel through the air at a speed of 2.997 x 108 m/s. Determine the wavelength of these radio waves. 2.930 m

29. Example Problem A transverse wave is observed to be moving along a lengthy rope. Adjacent crests are positioned 2.4 m apart. Exactly six crests are observed to move past a given point along the medium in 9.1 seconds. Determine the wavelength, frequency and speed of these waves. wavelength = 2.4 m speed = 1.6 m/s frequency = 0.66 Hz

30. Terms to Know Constructive interference- superposition of 2 or more waves in which individual displacements on the same side of the equilibrium position are added together to form the resultant wave. Destructive interference- a superposition of 2 or more waves in which individual displacements opposite sides of the equilibrium position are added together to form the resultant wave. Standing wave- a wave pattern that results when 2 waves of the same frequency, wavelength, and amplitude travel in opposite directions and interfere.

31. Few More Terms Node- a point in a standing wave that maintains zero displacement Antinode- a point in a standing wave, halfway between two nodes, at which the largest displacement.

32. Constructive Interference If we add these two waves together, point-by- point, we end up with a new wave that looks pretty much like the original waves but its amplitude is larger. This situation, where the resultant wave is bigger than either of the two original, is called constructive interference. The waves are adding together to form a bigger wave.

33. Destructive Interference When the first wave is up, the second wave is down and the two add to zero. When the first wave is down and the second is up, they again add to zero. In fact, at all points the two waves exactly cancel each other out and there is no wave left! The sum of two waves can be less than either wave, alone, and can even be zero. This is called destructive interference.

34. Reflection Free boundary= waves are reflected Fixed boundary= waves are reflected and inverted This downward force on the rope causes a displacement in the direction opposite the displacement of the original pulse. As a result, the pulse is inverted after reflection.

35. Standing Wave Node- a point in a standing wave that maintains zero displacement Antinode- a point in a standing wave, halfway between two nodes, at which the largest displacement.