Emergencies: eClass: 1-780-492-9372 Professor Mann: imann@ualberta.ca Simple Harmonic Oscillators (SHO) Basics πΉπ = −ππ₯ π=√ π= - Phase is ππ‘ + π - π is independent of π΄ - If an equation in the form π₯Μ = π2 π₯, then SHO - Remember that π, π, π, & π are heavily related! - Max force when cos = 1 ∴ πΉmax = ππ2 π΄ (from F=ma & a(t)) - Amplitude and Phase depend on how system set into motion Initial Conditions: π = 2ππ π π π = 2π√ 1 π √ 2π π π= π£0 2 π΄ = √π₯02 + ( ) π π£0 π = tan−1 (− ) ππ₯0 π π 1 π Displacement, velocity, accel π₯(π‘) = π΄πππ (ππ‘ + π) π π£(π‘) = ππ΄πππ (ππ‘ + π + ) 2 π£(π‘) = −ππ΄π ππ(ππ‘ + π) π(π‘) = π2 π΄πππ (ππ‘ + π + π) π(π‘) = −π2 π΄πππ (ππ‘ + π) Energy 1 πΈπ = ππ2 π΄2 sin2 (ππ‘ + π) 2 1 πΈπ = ππ΄2 cos 2(ππ‘ + π) 2 1 1 πΈπ‘ππ‘ππ = ππ΄2 = ππ2 π΄2 2 2 - π affects amplitude of velocity & acceleration equations. π - Since π = √ , that means k & m affect amplitude also. π - Read SHM graphs as sweep bar! - Energy “see-saws” between potential & kinetic - Total energy constant in undamped oscillator -πΈπ ∝ π₯ 2 Damped & Driven Simple Harmonic Oscillators (DSHO) Overdamped (π» > π): Damped SHO π0 = √ π₯(π‘) = π π π −π0 ππ‘ 2 (π΄π π0√π −1π‘ + 2 π΅π −π0√π −1π‘ ) Underdamped (π» < π): π 2√ππ 2ππ0 π₯Μ + 2π0 ππ₯Μ + π02 π₯ = 0 π π π₯Μ + π₯Μ + π₯ = 0 π π π = −π0 π ± π0 √π 2 − 1 πΉπ = −ππ₯Μ π₯(π‘) = π΄π −ω0 ππ‘ cos (π0 √1 − π 2 π‘ + π) Resonance & IP: Driven Oscillators: π= π = ππ = π0 √1 − 2π 2 πΉππ πππ ππππππ: π < 1 √2 ππ = π0 √2√1 − π 2 − 1 √3 πΉππ πππππππ‘πππ: π < 2 ππ = π0 √1 − π 2 Envelope: π΄π −π0ππ‘ π΄max = π΄0 π −π0 ππ‘ Period: 2π π= π0 √1 − π 2 Critically Damped (π» = π): π₯(π‘) = π −π0π‘ (π΄ + π΅π‘) π₯Μ (π‘) = π΅π −π0π‘ − π0 (π΄ + π΅π‘)π −π0π‘ π₯Μ + 2π0 ππ₯Μ + π02 π₯ = πΉ0 cos (ππ‘) π Steady state: πΉ0 1 π΄= ( ) π √(π02 − π 2 )2 + 4π02 π 2 π 2 π = tan−1 ( 2ππ0 π ) π02 − π 2 - If an equation is in the form π₯Μ + π₯Μ + π₯ = 0, then it’s a DSHO - Envelope & Period equations ONLY for underdamped - Critically damped returns to EQ position fastest. This also means it loses energy the fastest. - In underdamped systems, πΈ ∝ π΄π −π0ππ‘ - π is the “constant of proportionality” or “damping coefficient” (kg/s) - π is unitless. - When π = 1 & t = 0, π₯(0) = π΄, π΅ = π0 π₯(0) - ππ ≠ π0 - To find A and B for critical, set time to zero for displacement & velocity equations. (π΄ = π΄0 , π΅ = π0 π΄) - Taking the derivative o damping equations NOT hard. - Envelope traces “max amplitude” - As π ↑, ππ ↓ & less effect at resonance. - As π approaches 0, ππ approaches π0 - As π ↑, IP becomes less sudden/pronounced & ππ ↓. - Resonance & IP NOT tied together (but often close π) - Phase difference b/c of driving frequency: π΄π π = 0 ⇒ π π = 0, π΄π π = π0 ⇒ π = , π΄π π → ∞ ⇒ π = π 2 - After transient solution gone, π of driving oscillator is π of driven. - πΉ0 is amplitude of driving force Miscellaneous Uniform Circular Motion π π£ 2ππ = = π = π‘ π π = ππ‘ + π π₯ = ππππ (ππ‘ + π) π¦ = ππ ππ(ππ‘ + π) Dealing w/ arctan: Check π₯0 and π£0 equations to see if signs on left side = signs on right. Use CAST to check sin/cos. Add pi if wrong solution. Pendulums π π=√ π π = 2π√ π π π(π‘) = π΄πππ (ππ‘ + π) 1 πΈπ = ππ2 π 2 π΄2 sin2 (ππ‘ + π) 2 1 πΈπ = ππππ΄2 cos 2 (ππ‘ + π) 2 1 1 πΈπ‘ππ‘ππ = ππππ΄2 = ππ2 π 2 π΄2 2 2 - Pendulum formulas only true for π ≤ 10° - Small angle approximation necessary b/c no actual equation for pendulums - Underdamped equation also works on pendulums - πΈ ∝ π΄2 Wave Basics Basics 2π π= π π = ππ = π π π Δπ‘ = π Density: π π= π Pressure at a given point: πΉ πΏπΉ⊥ ππΉ⊥ π= (= lim = ) πΏπ΄→0 πΏπ΄ π΄ ππ΄ String-related π£=√ π ππΏ =√ πβ π πβ = π πΏ Wave Equations π 2 Ψ 1 ∂2 Ψ = ∂Ψ 2 c 2 ∂Ψ 2 Positive (moving right) π = −ππ₯ − π Ψ(π₯, π‘) = π΄πππ (ππ₯ − ππ‘ + π) Negative (moving left) ξ = ππ₯ + π Ψ(π₯, π‘) = π΄πππ (ππ₯ + ππ‘ + π) - Diff. b/t a wave & an oscillator is addition of length. - Period is time b/t crests & time for point to oscillate once - Partial DV’s are DV’s where 1 variable constant. - 2 Velocities: Phase velocity (v. of wave) & Particle velocity (v. of particle going up & down) - Wave #(k): # of cycles wave completes / dist. travelled. - Ψ is just fancy variable for “wave displacement” - t constant, “snapshot”. x constant, “point oscillating”. Variable on “x-axis” of graph is NOT constant. - Pressure constant throughout a fluid Acoustic Waves Equations π2π π π2π π2π π π2π = = ππ₯ 2 π΅ ππ‘ 2 ππ₯ 2 π΅ ππ‘ 2 - πβ is linear mass density. (kg/m), T is tension force (N) - y(x, t) is just displacement EQ applied to string. It’s DV, π£π¦ , is the transverse velocity of wave on string. - Variable c often speed of light. Here, it’s phase velocity -Think of y vs t graph as “sweep bar” and y vs x graph as “wave moving across” (reverse sweep bar when time) - ππ¦ = vertical component of tension force π΅ π Δππ0 ππ2 = 2 Δπ π π¦(π₯, π‘) = π¦π sin(ππ₯ ± ππ‘ + π) π£π¦ = −ππ¦π cos(ππ₯ − ππ‘ + π) ππ¦ π£π¦ = | ππ‘ π₯ Basic power formula: π = ππ¦ π£ Instantaneous Power: π=√ π = √πππ2 π΄2 sin2 (ππ₯ − ππ‘ + π) π π(π₯, π‘) = π΅ππ΄Ψ cos (ππ₯ − ππ‘ + π + ) 2 Average power: 1 π = √πππ2 π΄2 2 π΅=− Displacement: Ψ(π₯, π‘) = π΄Ψ cos(ππ₯ − ππ‘ + π) Pressure: πΨ π(π₯, π‘) = −π΅ ππ₯ π(π₯, π‘) = π΅ππ΄Ψ sin(ππ₯ − ππ‘ + π) π΄π = π΅ππ΄Ψ = ππππ΄Ψ - Bulk modulus (B) is measure of incompressibility. Δ(ππ’ππ π π‘πππ π ) ππππ π π’ππ πβππππ - π΅ = Δ(ππ’ππ π π‘ππππ) = π£πππ’ππ πβππππ Bulk = in all directions - Fluids ONLY allow longitudinal waves (acoustic) π - Displacement is behind pressure. 2 - Ψ relative to EQ pos, π relative to room p (p EQ pos) πΨ ∝ π. High B means high Ψ needed to affect p. ∂x Interference, Power, & Intensity Interference (Assuming same phase waves) Constructive Interference: Ψ(π₯0 , π‘) = (π΄1 + π΄2 ) cos(ππ₯0 − ππ‘) ππ π Destructive Interference: sin(π) = Ψ(π₯0 , π‘) = (π΄1 − π΄2 )cos (ππ₯0 − ππ‘) 1 (π + ) π 2 sin(π) = π Diff. in phase from length: Δπ = πΔπ Combined waves on string: π π π¦πππ‘ (π₯, π‘) = [2π¦π cos ( )] sin (ππ₯ − ππ‘ + ) 2 2 Power & Intensity πΈ π π= πΌ= π‘ π΄ Mean intensity: 1 1 πΌ = π΅πππ΄2 = √π΅ππ2 π΄2 2 2 Mean I at distance (3D&2D) π π π π πΌ= ( ) πΌ= ( ) 4ππ 2 π2 2ππ π Decibels: πΌ π½ = 10ππ΅ β log ( ) πΌ0 πΌ1 Δπ½ = π½1 − π½2 = 10ππ΅ β log ( ) πΌ2 - Superposition: Overlapping waves can be summed. - Boundaries: Fixed: Ψ = 0, πππ£πππ‘. Free: π ππππ = 0 - Waves same phase: interference due to source location - If Δπ = ππ : destructive. If π2π: constructive - If ΔπΏ = ππ 2 βΆ destructive. If ππ βΆ constructive - Energy (any form, power, intensity, etc) ∝ π΄2 - Sound is isotropic (spreads uniformly in all directions) - Emitting often about power. Receiving about intensity - πΌ0 = threshold of audibility. 10−12 π/π2 - π½ = 0 → πΌ = πΌ0 . π½ > 0 → πΌ > πΌ0 . π½ < 0 → πΌ < πΌ0 π - Combined waves on string: 2π¦π cos ( ) is amplitude. 2 Beats Avg. angular freq. (π Μ ) 1 π Μ = (π1 + π2 ) 2 Diff. in angular freq. (Δπ) Δπ = π1 − π2 1 1 π1 = π Μ + Δπ π2 = π Μ − Δπ 2 2 Resulting displacement: - “Amplitude” of beat wave is modulated by ω1 − π2 ω1 + ω2 Ψ(π‘) = 2π΄πππ ( π‘) cos ( π‘) 2 2 2π΄πππ ( Envelop wave & Beat Frequency: π1 − π2 ππΈ = ππ = |π1 − π2 | 2 - Amplitude is also envelope wave EQ, while perceived frequency is frequency EQ - Beats are 2x envelope wave. π1 −π2 2 π‘) while frequency is cos ( π1 +π2 2 π‘) Trig Identities Basic Identities Sine Identities sin(π₯) = − sin(−π₯) π sin(π₯) = cos ( − π₯) 2 sin(π₯) = sin(π − π₯) sin(π₯) = − sin(π + π₯) sin(π₯) = sin(2π + π₯) Cosine Identities cos(π₯) = cos(−π₯) π cos(π₯) = sin ( − π₯) 2 cos(π₯) = − cos(π − π₯) cos(π₯) = − cos(π + π₯) cos(π₯) = cos(2π + π₯) Tangent Identities tan(π₯) = − tan(−π₯) 1 tan(π₯) = π tan ( − π₯) 2 tan(π₯) = − tan(π − π₯) tan(π₯) = tan(π + π₯) Key Concepts • Pythagorean Identities 2 2 (π) cos (π) + sin =1 2 (θ) 2 1 + tan = sec (θ) 1 + cot 2 (θ) = csc 2 (θ) Sum & Difference Identities sin(π + π) = sin(π) cos(π) + sin(π) cos(π) sin(π − π) = sin(π) cos(π) − sin(π) cos(π) cos(π + π) = cos(π) cos(π) − sin(π) sin(π) cos(π − π) = cos(π) cos(π) + sin(π) sin(π) tan(a) + tan(b) tan(π + π) = 1 − tan(a) tan(b) tan(π) − tan(π) tan(π − π) = 1 + tan(π) tan(b) Double Angle Identities Derived Identities cos(π − π) − cos(π + π) 2 cos(π + π) + cos(π − π) cos(π) cos(π) = 2 sin(π + π) + sin(π − π) sin(π) cos(π) = 2 π+π π−π sin(π) + sin(π) = 2 sin ( ) cos ( ) 2 2 π−π π+π sin(π) − sin(π) = 2 sin ( ) cos ( ) 2 2 π+π π−π cos(π) + cos(π) = 2 cos ( ) cos ( ) 2 2 π+π π−π cos(π) − cos(π) = −2 sin ( ) sin ( ) 2 2 sin(π) sin(π) = sin(2π) = 2 sin(π) cos(π) cos(2π) = cos 2 (π) − sin2 (π) cos(2π) = 1 − 2 sin2 (π) cos(2π) = 1 + 2 cos 2(π) 2 tan(a) tan(2π) = 1 − tan2 (π) Know what variables you’re looking for, then see which equations involve them. Those equations may be what is needed to find your missing variable!