Lecture Notes- Pendulums, Waves, Sound

Pendulums and Oscillations

• A pendulum consists of a mass at the end of a string.
• The time for one back-and-forth oscillation is fixed, depending on the length of the string.
• This is called simple harmonic motion or periodic motion.
• The main factor affecting the pendulum's motion is its length.

Frequency and Period

• Frequency: The number of oscillations in a certain amount of time.
• Units of frequency: Hertz (Hz).
• Period ($T$): The time for one oscillation.
• T = arc{1}{frequency}
• If the frequency is high (many oscillations per second), the period is very small.

Amplitude

• Amplitude: How far the pendulum goes away from the center.
• Usually, it oscillates the same amount on either side of the center.
• Measurement is taken from the center to one extreme.

Hooke's Law and Periodic Motion

• If an object follows Hooke's law, it will undergo periodic motion.
• Hooke's law states that the restoring force is proportional to the displacement.
• $F = -kx$ (where $F$ is the force, $k$ is the spring constant, and $x$ is the displacement)
• The force is proportional to the change in position (displacement).
• The farther from the center, the greater the restoring force.
• As long as the angle is small, the pendulum follows this relationship.

Displacement and Restoring Force

• Displacement: How far an object is from the center.
• For Hooke's law, if displacement is away from the center, the restoring force is back towards the center.
• The force is proportional to the displacement but in the opposite direction.
• Supporting a system at the top (like a pendulum) results in periodic motion.
• Supporting a system at the bottom leads to it simply falling over.

Waves

Wave Motion

• In a wave, oscillation extends through space.
• There is motion of the wave and motion of the material that makes up the wave.

Transverse Waves

• In a transverse wave, the wave and the medium move at right angles.
• Example: A sports wave in a stadium, where fans move up and down as the wave moves around.

Longitudinal Waves

• In a longitudinal wave, the wave and the medium move in the same direction.
• Example: Traffic waves, where cars move forward and the wave of braking propagates backward.

Wavelength, Frequency, and Wave Speed

• Wavelength: The distance that the wave travels in one oscillation.
• Wave speed ($v$) can be calculated using wavelength ($\lambda$) and period ($T$).
• v = arc{\lambda}{T}
• Since f = arc{1}{T}, the formula becomes $v = \lambda f$

Doppler Effect

• The Doppler effect is a change in frequency due to motion.
• If a bug is moving across the water, the waves get closer together in the direction of motion (higher frequency) and farther apart in the opposite direction (lower frequency).

Shock Waves

• If an object is traveling at the speed of the wave, all the waves pile up.
• If an object goes faster than the wave speed, energy dissipates behind it as a shockwave.
• Examples: A plane traveling faster than the speed of sound or the crack of a whip.

Sound Waves

Sound Transmission

• Sound waves are vibrations that travel through a medium, usually air.
• Sound involves compression (increase in pressure) and rarefaction (decrease in pressure) of the medium.
• No sound in space because there is no medium to compress and rarefy.

Audible, Infrasonic, and Ultrasonic Sound

• Audible sound: Sound that humans can hear (approximately 20 Hz to 20,000 Hz).
• Infrasonic: Frequencies below 20 Hz (inaudible).
• Ultrasonic: Frequencies above 20,000 Hz (inaudible).

Pitch and Echoes

• Pitch: The perception of sound frequency by the brain.
• Echo: Reflection of sound.
• Reverberation: Multiple reflections of sound.

Reflection and Refraction of Sound

• Reflection: Sound bounces off a surface at the same angle it hits the surface.
• Refraction: Bending of a wave as it passes from one medium to another.
• Refraction occurs because the speed of sound is different in different materials.
• Angles are measured from the normal (perpendicular line to the surface).
• Ultrasonic imaging uses reflections and refractions to reconstruct images of the inside of the body.

Ultrasonic Imaging

• Ultrasonic imaging uses high-frequency sound waves to see small details.
• It's safer than surgery.
• High frequency = short wavelength, therefore smaller the wave length, the better it is at seeing small details.
• The ultrasound frequency is outside hearing range, so we don't damage our ears.

Interference

• Interference: When two or more waves overlap.
• Constructive interference: Waves are in phase (up at the same time, down at the same time), resulting in a larger amplitude.
• Destructive interference: Waves are opposite each other, resulting in cancellation.
• Noise-canceling headphones use destructive interference.

Standing Waves

• Standing wave: A wave interfering with itself, usually due to reflection.
• In a string of a particular length, only certain waves will create constructive interference.
• These waves are multiples of half wavelengths.

Harmonics

• First harmonic (fundamental frequency): The lowest frequency.
• Higher harmonics are multiples of the fundamental frequency (e.g., second harmonic is twice the first).
• Different instruments sound different because of the different ratios of harmonics.
• A piano playing a C and a trumpet playing a C sound different because of these different ratios.

Instruments

• The different harmonics have different amounts of energy in them. One instrument might have all of its energy in just the first harmonic but another one might have a lot of energy in the even harmonics or it might have a lot of energy in the odd harmonics.
• String instruments and wind instruments have different ways of producing oscillations.
• Instruments amplify particular frequencies.

Sounding Boards

• Sounding board: Amplifies the sound.
• The body of a guitar is a sounding board that vibrates and moves more air.
• A harp does not have a sounding board, so the oscillations last longer but are quieter.

Resonance

• Resonance: When the frequency of the source of sound matches the frequency of the object, energy is transferred quickly.
• Example: A crystal glass shattering when someone sings at its resonance frequency.
• Bridges can collapse due to resonance frequencies matching wind frequencies (e.g., the Tacoma Narrows Bridge collapse).

Beat Patterns

• Beat pattern: When two waves of similar frequency interfere.
• The frequency of the beat depends on the difference between the frequencies of the two waves.
• This is used to tune instruments.
• When the instrument is in tune, you hear a steady note. If the instrument is out of tune, depending on how far they're out of tune, will affect how quickly it alternates between the constructive and destructive.

Practice Test Questions

1. Momentum: Truck parked, but a parked building will always have zero velocity, hence, no momentum.
2. Impulse: $Impulse = Force * Time$. Impulse is force multiplied by time.
3. Gun: Target safer because lighter gun moves backward quickly because the bullet is more massive than the gun.
4. Falling: Momentum is greater with the more massive object when falling. The heavier one will have a greater momentum.
5. Knees: Average force greatly decreases.
6. Apple: Force depends on all given factors.
7. Putty: Momentum is conserved before and after collision ($p = mv$).
8. Rocket: Force exerted by exhaust gases.
9. Cannon: Longer barrel increases muzzle velocity due to prolonged pressure.
10. Shark and Fish: $m<em>1v</em>1 + m<em>2v</em>2 = (m<em>1 + m</em>2)v_f$; resulting speed is positive (0.17 m/s).
11. Freight car: Inelastic collision; use conservation of momentum to find initial speed.
12. Wall: Work done is zero because distance is zero (Work = Force * Distance).
13. Work: Power remains the same (Power = Work/Time).
14. Potential Energy: Doubles when height doubles ($PE = mgh$).
15. Location: Potential energy depends on location.
16. Airplane: Momentum decreases when bullets are fired forward.
17. Perpetual: Always false; you cannot create energy.
18. Inclined Marble: Half converts to KE; they are the same.
19. Bowling ball: Angle is not being converted.
20. Car speed: Faster car has 16 times the energy.
21. Jack efficiency: Always equal to 50% because $eff = \frac{workOutput}{workInput}$.
22. Car breaking: Four times as far.
23. Work: time is force multiplied by distance.
24. Kinetic Energy: momentum is directly proportional to energy.
25. Rifle: Bullet has more speed and KE due to less mass with consistent momentum.
26. Merry go round: Farthest from the center has the largest linear speed.
27. Tires: Because you are having fewer rotations for the same distance, the readings will decrease.
28. Coin: Easier to rotate. The coin has some mass near the center.
29. Spider monkeys: They easily vary to change their center of mass.
30. Tower Pizza: Decreases!
31. Seesaw: rises!
32. Jupiter: rotation increases!
33. Transverse: At a right angle.
34. Frequency: It depends on length, and on gravity.
35. Seconds: Divide the time given by 100.
36. Wave: The same speed!
37. Waves: F*W= 20, since W =10.
38. Waves: You are getting a lower frequency.
39. Energy: Amplitude!
40. Sound: Always Steel!!!
41. Speed of sound: Travels faster in a warmer medium!
42. Fraction: Change in the speed of wave!
43. Air eventually becomes:
44. Bass: The answer is a sounding board!
45. The crystal chandelier: Resonance!
46. radio AM: Amplitude!
47. Hot: The waves refract downward!
48. Helium: The medium around the voice travels faster!