Physics 20 - Oscillatory Motion & Mechanical Waves Notes

Mechanical Waves

• Mechanical waves transmit energy through a medium via vibrations.
• We'll explore the nature, properties, and energy transmission of waves.

8.1: Properties of Waves

• Wave: A pattern traveling through a medium via vibration, an oscillation carrying energy.
• In a perfectly elastic medium, waves may exhibit Simple Harmonic Motion (SHM).

Terminology:

• Wave Front: An imaginary line connecting points reached by a wave simultaneously.
• Incident Wave: A wave front moving from the origin towards a barrier.
• Reflected Wave: A wave front moving away from a barrier.
• Wave Train: A continuous series of crests and troughs.
• Point Source: A single disturbance point generating a circular wave.
• Ray: Shows the direction of wave front motion.

Waves and Rays

• Waves move at right angles to the crest line, indicated by a velocity vector or ray.
• Ripple Tank Example
• Diverging Rays: Spread out from the origin, indicating energy dispersal over a growing area.

Wave Properties

• Amplitude: Maximum displacement from equilibrium.
• Wavelength: Distance between two identical points on a wave (crest to crest, trough to trough), where the wave begins to repeat its motion.

Waves and Boundaries: Reflection

• When a wave meets a boundary, energy is transferred and reflected.

Example 1:

• A wave front hits a barrier at a $30^(circ$ angle.
• The angle of incidence equals the angle of reflection: $\theta<em>i = \theta</em>r$.

8.4: The Doppler Effect

• Occurs when a wave source (sound, light) is moving.

Doppler Equation:

• $f<em>d = (\frac{v</em>w}{v<em>w \pm v</em>s}) f_s$
• $f_d$ = Doppler frequency (apparent frequency).
• $v_w$ = Speed of sound in air (approximately $340 m/s$).
• $v_s$ = Speed of source.
• $f_s$ = True frequency of source.
• Use minus (-) when the source moves toward the observer.
• Use plus (+) when the source moves away from the observer.

Example 1:

• A train travels at $30.0 m/s$, its whistle emits sound at $224 Hz$.
• Calculate the frequency change as the train passes.

Bow Waves

• Bow wave forms at the bow of a boat moving through water.
• It defines the outer limits of a boat's wake.
• For airplanes, the bow wave is a pressure wave traveling at the speed of sound.

The Sound Barrier

• Crests of bow waves arrive simultaneously, creating a sonic boom.
• Sonic boom: Sound from shock waves of an object exceeding the speed of sound; sounds like an explosion.
• The sound barrier is the transition point from the speed of sound to supersonic speed.

Chapter 7: Oscillatory Motion Conditions

• Oscillatory Motion: Motion with a constant period for each cycle.

Example 1:

• Find the frequency of an engine with a piston oscillating at a period of $0.0625 s$.

• $T = \frac{t}{cycles}$ (Period)

• $f = \frac{cycles}{s}$ (Frequency)

• $f = \frac{1}{T}$

• $f = \frac{1}{0.0625} = 16.0 Hz$

8.3: Superposition and Interference

• Waves interact momentarily, then continue unaffected.

Principle of Superposition:

• The displacement caused by multiple waves is the algebraic sum of individual amplitudes.
• Constructive Interference
• Destructive Interference

Standing Waves

• Observed on strings fixed at both ends.
• Resonant frequencies: Frequencies at which standing waves occur for a given string length.
• Incident and reflected waves interfere.
• Constructive interference (antinodes) where crests meet crests, troughs meet troughs.
• Destructive interference (nodes) where crests meet troughs.
• Nodes: Points with minimal displacement.
• Antinodes: Points with maximum displacement.

Resonating Air Columns

• Wind instruments use resonance for sound production.

Closed Columns:

• Node at the closed end.
• Antinode at the open end.
• $l = \frac{1}{4} \lambda$
• $\lambda = 4l$
• $v = f \lambda$
• $f = \frac{v}{\lambda}$
• Fundamental Frequency/1st Harmonic: $f = \frac{v}{4l}$
• Only odd harmonics produce sound in closed columns.

Open Columns:

• Antinodes at both ends.
• $l = \frac{\lambda}{2}$
• $\lambda = 2l$
• Fundamental Frequency/1st Harmonic: $f = \frac{v}{2l}$
• All harmonics produce sound in open columns.
• $v$ is the speed of sound ($340 m/s$).

Example 1:

• Open-ended organ column length $l = 3.6 m$.

• Determine the wavelength of the fundamental frequency: $\lambda = 2l = 2(3.6) = 7.2 m$.

• Determine the frequency: $f = \frac{v}{\lambda} = \frac{346}{7.2} = 48 Hz$.

• The third harmonic: $f = 3(\frac{v}{2l}) = 3(\frac{346}{2(3.6)}) = 144 Hz$

• Increasing column length lowers the fundamental frequency.

Interference Patterns

8.2: Transverse and Longitudinal Waves

Transverse Waves:

1. Classical wave shape.
2. Particles move perpendicular to wave direction.
3. Example: Ripples on a pond.

Longitudinal Waves:

1. Displacement parallel to wave propagation.
2. Series of compressions and rarefactions.
3. Example: Sound waves.

Waves, Energy, and the Medium

• Solids store elastic potential energy via longitudinal or transverse stretching.
• Fluids store elastic potential energy via compression; waves are pressure waves.
• Transverse waves in liquids are surface waves due to gravitational potential energy.

Pulse and the Universal Wave Equation

• Pulse: A single crest or trough; length is half the wavelength.
• Amplitude increases energy.
• Speed from Universal Wave Equation: $v = \frac{l}{\Delta t}$. When l equals lambda, we get $v = \frac{\lambda}{T}$, therefore we derive $v=\lambda f$
• $\lambda$ = Wavelength (m)
• $T$ = Period (s)
• $f$ = Frequency (Hz)

Example 1:

• Wrist flick creates a pulse in $0.54 s$ with a length of $0.78 m$.
• $v = \frac{l}{t} = \frac{0.78}{0.54} = 1.4 m/s$.

Example 2:

• Speed of light: $3.00 \times 10^8 m/s$.
• Wavelength of red light: $700 nm$.
• $f = \frac{v}{\lambda} = \frac{3.00 \times 10^8}{700 \times 10^{-9}} = 4.29 \times 10^{14} Hz$.

Waves and Boundaries

• Waves change velocity when moving between media.
• Constant source frequency causes changes in velocity and wavelength: $v = \lambda f$.

Change in Medium Density:

• Denser to less dense: Reflected the same way.
• Less dense to denser: Reflects inverted.

Change in Depth:

• Waves slow in shallow water, decreasing wavelength (constant frequency).

String Attached to…

• Fixed end: Inverted reflection.
• Loose end: Reflection is the same way.

7.2: Simple Harmonics

Hooke’s Law:

• Deformation is proportional to force.
• Spring deformation relates directly to applied force.
• Spring stiffness (spring constant) is represented by $k$.
• Equation: $F = kX$

Hooke’s Law Example 1:

• Plot force vs. displacement to find the spring constant.
• $k = \frac{\Delta F}{\Delta x} = 2 N/m$

The Restoring Force

• Spring exerts equal and opposite force.
• Restoring force tries to return the spring to equilibrium.
• Symbol: $F_r$.
• $F<em>a = -F</em>r$

Example 2:

• $k = 30.0 N/m$, $x = 1.50 m$.
• $F_r = -kx = -(30)(-1.5) = 45.0 N$

Simple Harmonic Motion (SHM) of Horizontal Mass-Spring System

• SHM: Motion about equilibrium caused by restoring force proportional to displacement.
• Horizontal mass-spring system oscillates with SHM.

Key Points:

• Equilibrium position:
  • $x = 0$, $F_r = 0$, $a = 0$, $v = max$
• Max displacement (+A):
  • $x = +max = +A$, $F_r = -max$, $a = -max$, $v = 0$
• Max displacement (-A):
  • $x = -max = -A$, $F_r = +max$, $a = +max$, $v = 0$

Simple Harmonic Motion of Vertical Mass-Spring System

• Similar to horizontal, but with gravitational force ($F_g$).

Example 3:

• Mass of $510 g$ stretches spring $0.500 m$.

Simple Harmonic Motion of a Pendulum (small angles)

• Exhibits SHM for small displacements.

Motion with Large Amplitudes

• Non-linear graph; restoring force not proportional to displacement.

Example 4:

• Find the restoring force for a pendulum bob of mass $100.0 g$ at an angle of $10.0^\circ$.

7.3: Position, Velocity, Acceleration & Time Relationships

Acceleration of a Mass-Spring System

• $F<em>{net}$ and $F</em>s$ are used to describe the force.
• $a = -(\frac{k}{m})x$ (Instantaneous acceleration)
• Maximum acceleration occurs at maximum displacement (A).
• Acceleration decreases as it approaches equilibrium.
• Acceleration and displacement are in opposite directions.

Maximum Speed of a Mass-Spring System

• Total mechanical energy is constant in an isolated system.
• $E<em>m = E</em>k + E_p$
• $E<em>{pmax} = E</em>{kmax}$
• $\frac{1}{2} kA^2 = \frac{1}{2} mV_m^2$

Example 1:

• $100.0 g$ mass, $k = 1.014 N/m$, displacement of $15.0 cm$ [up]: $a = -(\frac{k}{m})x$

Period of A Mass-Spring System

• Circular motion (CM) describes Simple Harmonic Motion (SHM).

Three Conditions:

1. Radius (r) of CM equals amplitude of SHM.
2. Mass in CM with constant speed.
3. Periods of CM and SHM are the same.

• Equation for Period:
  • $T = 2 \pi \sqrt{\frac{m}{k}}$
• Maximum speed of mass-spring system = speed of CM.

Example 2:

• Amplitude is $12.25 cm$, maximum speed is $5.13 m/s$, $k = 5.03 N/m$

The Period of A Pendulum (angles 15o or less)

• Pendulum's mass does not affect the period.

The Pendulum and Gravitational Field Strength

1. g varies with altitude and latitude.
2. $g = 9.81 N/kg$ at Earth's average radius.
3. Pendulums determine exact $g$ using the period formula.

The Pendulum's Period Equation Formula.

• $T = 2\pi\sqrt{\frac{L}{g}}$

Example 3:

• Find gravitational field strength at Mount Everest (altitude $8954.0 m$), pendulum length $1.00 m$, period $2.01 s$.