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Physics final notes from textbook

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McMaster University
Reza Nejat

Chapter 15- Oscillatory Motion • Simple Harmonic Motion: an object moves with SHM when its acceleration is proportional to its position and is oppositely directed to the displacement from equilibrium o A x -(k/m)(x) o Speed is maximum whenA=0 since acceleration changes signs o Acceleration is maximum whenA= positive or negative X and speed is zero max o Block oscillates between its turning point x= positive and negative x • Particle in SHM o ω = k/m o position of particle as a function of time  x (t) =Acos (ωt+ϕ) • A, ω, ϕ are constant • A= amplitude of the motion; max value of particle in the positive or negative direction • ω = angular frequency with units radians per second; measure of how frequently the oscillation are occurring; the greater the oscillations per unit time, the greater value of ω k/m o ω = √ • ϕ = phase constant (or initial phase angle); determined by the position and velocity of particle at t=0 o to get phase constant, set t=0 and solve for v(0) and x(0) and then divide the eqn from each other; find phase constant, then amplitude from the x(0) eqn and then v and max and then the 3 eqn for position, velocity and acceleration o if particle is at max position x=Aat t=0, then ϕ= 0 • (ωt+ϕ)= phase of motion  Period T= time interval required for the particle to go through one full cycle of its motion • T= 2π/ ω = 2π/ √ k/m or 2π* √ m/k ; measures seconds per cycle • f= 1/T=1/2π/ ω= ω/2π = [( √ k/m )/2π] ;units are Hz and measures cycles per second • period and frequency only depend on the mass of the object and the force constant of the spring and notAor phase constant  v= -ωA sin (ωt+ϕ) and v max/minv (plus or minus) ωA  a= -ω Acos (ωt+ϕ) and a max/mina (plus or minus) ω A  x and v differ in phase by 90 degrees and when x is max or min, v=0 and when x=0 v= max or min  x and a differ in phase by 180 degrees so when x is max, a is max in opposite direction  when block moves to right, the ϕ is negative and when block moves to left, ϕ is positive  period is independent of how the oscillator is set into motion (i.e initial velocity) • Comparing SHM with Uniform Circular Motion o Angular speed of the particle in uniform circular motion (UCM) is same as angular frequency ω o The initial angle particle in UCM makes with x axis is same as phase constant ϕ o The radius of particle in UCM is the same as amplitude o Speed of particle in UCM is v=rω which is same as speed of particle ωA 2 o Acceleration of particle in UCM is same as acceleration of particle ω A • The Pendulum o Exhibits periodic motion; consists of particle-like bob of mass, m and is suspended by a light spring of length, L; motion occurs in vertical plane and is driven by gravitational force o Provided that angle is small (less than 10 degrees) between the string and the vertical plane, the motion is very close to SHM o mgsinϴ is the restoring force heading towards the equilibrium of 0 degrees and -mgsinϴ means that the restoring force is opposite the displacement o if we assume that angle is small, then we can use small angle approximation where sinϴ=ϴ and assume than pendulum follows SHM  velocity= -(g/L)ϴ (whereϴ is smaller than 10 degrees)  ω= √g/L angular frequency for simple pendulum  T = 2π/ω = 2π* √ L/g period of simple pendulum • Period and frequency of simple pendulum depend only on length of the string and acceleration on gravity; period is independent of mass so all simple pendula of equal length and at same location (for constant g) on x axis have the same period • Energy of Simple Harmonic Motion 2 2 2 2 o K=0.5mv = 0.5 m (ω A sin (ωt+ϕ)) Kinetic energy of a simple harmonic oscillator (the block) 2 2 2 o U= 0.5 k x = 0.5 k (A cos (ωt+ϕ)) Potential energy of simple harmonic oscillator 2 2 2 2 o E= K+U = 0.5 kA (cos (ωt+ϕ)) + sin (ωt+ϕ)) note: ω =k/m so that is taken out as a factor and since cos (ωt+ϕ)) + sin (ωt+ϕ)=1 so E=0.5 kA 2  this is the total energy of a simple harmonic oscillator and is equal to maximum potential energy of a spring at x=positive or negative x since velocity=0 at that point o Velocity of block at an arbitrary position when both K and U exist A −x 2 2 2  v= ± k/m(¿)=±ω A −√ √¿ -------------------------------------------------------------------------------------------------------------------- - Chapter 16-Waves • Propagation of Disturbance o Wave motion is the transfer of energy through space without the accompanying transfer of matter o Mechanical waves and electromagnetic radiations depend on waves o All mechanical waves require: (e.g. a string under tension with hand flicking the string)  Source of disturbance- hand  Amedium containing elements that can be disturbed-string  Some physical mechanism through which elements of the medium can influence each other-the elements of the string are connected to one another so influence each other o Wave: periodic disturbance travelling through a medium o Transverse wave- a travelling wave or a pulse that causes the elements of the disturbed medium to move perpendicular to the direction of propagation  a string under tension with hand flicking the string o Longitudinal wave- a travelling wave or a pulse that causes the elements of the disturbed medium to move parallel to the direction of propagation  Stretching and compressing coil with one end attached to wall and parallel to ground o Vertical position of an element of the medium at any point, P is given by  Y= f(x+vt) (movement to left) and Y= f(x-vt) (movement to right) • This function depends on both x and t thus written y(x,t)  Y(x,t) represents the y coordinate-the transverse position- or any element located at position x at any t; the element at position x moves up, gets to max and then decline once the wave had passed (the vertical position of a particle located at x at time t when the wave passes through it) • The Speed of Waves on Strings o speed of a transverse pulse travelling on a taut string o acceleration of element increases with increasing tension and so wave speed increases with increasing tension o likewise, because it is more difficult to accelerate an element of a massive string than that of a light string, the wave speed should decrease as the mass per unit length increases  light string and high tension= greater wave speed  v= √T/μ Speed of a wave on a stretched string  The wave speed is determined by the medium, so it is unaffected by changing the frequency • Analysis Model: Travelling Wave o motion of elements of the medium is in SHM, moving up and down o motion of the wave is right or left o analysis model of a traveling wave  used where wave moves through space without interacting with other waves o crest is the displacement of the element from its normal position and lowest point is called the trough o wavelength is minimum distance between any 2 identical points on adjacent waves o period is the time interval required for 2 identical points of adjacent waves to pass by a point o frequency of a periodic wave if the number of crest (or trough or any other point on wave) that pass a given point in a unit time interval; units= Hz  f=1/T o amplitude is the maximum position of an element of a medium relative to its equilibrium position o speed of wave depends on the properties of the medium o function describing the position of the element of the medium through which the sinusoidal wave is travelling through is: 2π (x)¿  y(x,0)=Asin[ λ  at any given time t, y has the same value at the position x, x + λ, x+2λ o if the wave move to the right with a speed, v, the wave function at some later time, t is: 2π (x−vt)¿  y(x,t)=Asin[ λ , if wave travelling to left then replace – with + o angular wave number 2π  k= λ , in rad/m o angular frequency  ω=2π/T=2πf , in rad/s o wave function for sinusoidal wave  y=Asin (kx-ωt) , this is when y=0 when x=0 and t=0 (and so ϕ=0)  y=Asin (kx- ωt+ ϕ), this is the general case when y=0 when x=0 and t=0 is not the case • determine ϕ by setting t=0 and x=0 (initial conditions) o Speed of a sinusoidal wave ω  v = λ/T = fλ = k o transverse speed (speed of element) and transverse acceleration respectively (when y=0 when x=0 and t=0)- note: either x or t must be held constant since y depends on both so this is the speed of the element when either the particle is at a constant position x or at a constant time t  vy= -ωA sin (kx-ωt) and v max/min v ±ωA • reaches max value when y=0  ay=-ω Acos (kx-ωt) and a max/mina ±ω A • reach max value when y is ±A • Rate of Energy Transfer by Sinusoidal Waves on String o Waves transfer energy through a medium as they propagate o The element moves in SHM in the y direction o Power of a wave  P=1/2 µω A (λ/T)= 1/2 µω A v and units are watts  this is the rate of energy transfer by a sinusoidal wave on a string which is proportional to the square of the frequency, the square of the amplitude and the wave speed o Each element can be considered to have a mass of dm o This gives a total energy of  El= Kl+ Ul = ½μ w A λ 2 • Reflection and Transmission o How a traveling wave is affected when it interacts with a change in medium o When a wave traveling on a taut string reaches the fixed-end (hits a wall), it reflects  Reflection- the pulse moves back along the spring in the opposite direction  Reflected pulse is inverted o When a wave traveling on a taut string reaches the free-end , it reflects but is NOT inverted o When a wave traveling on a taut string reaches a boundary intermediate between the free-end and fixed-end, part of the energy in the incident pulse is reflected and part undergoes transmission, that is, some of the energy passes through the boundary  E.g. light string is attached to a heavier string  Light string pulse, when reaches heavier string, part is reflected while part energy goes into the heavier string • speed of wave is faster and then becomes slower • the reflected pulse is inverted since the heavier string is analogous to the fixed end • the reflected pulse has a smaller amplitude as compared with the incident pulse (energy carried by wave is related to its amplitude) • according to principal of conservation of energy, when the pulse breaks into reflected and transmitted wave, the sum of the energies must equal to the energy of the incident pulse  heavier string pulse, when reaches lighter string, part is reflected while part energy goes into the lighter string • speed of wave is slower and then becomes faster • the reflected pulse is not inverted since the lighter string is analogous to the free end  the relative heights of the reflected and transmitted pulse depends on the relative densities of the 2 medium • if same densities then no reflection  lighter string means faster wave speed if tension is remains constant -------------------------------------------------------------------------------------------------------------------- - Chapter 18- Superposition • Analysis Model: Waves in Interference o this is combination of traveling waves o superposition principle  If two or more waves are travelling through a medium, the resultant value of the wave function at any point is the algebraic sum of the values of the wave function of the individual waves. • Two travelling waves can pass through each other without being destroyed or altered  waves that obey this principle are called linear waves • have amplitudes much smaller than their wavelength  waves that do not obey this principle are called non-linear waves • have amplitudes that are much larger o interference: combination of separate waves in the same region of space to produce a resultant wave o constructive interference: displacement caused by 2 pulses are in the same direction (2 waves heading towards each other but have their pulses with same y direction element displacement) o destructive interference: displacement caused by 2 pulses are in the opposite direction (2 waves heading towards each other but have their pulses with different y direction element displacement) o superposition principle in 2 sinusoidal waves travelling in the same direction in a linear medium (have same frequency, wavelength, amplitude but differ in phase)  y 1Asin (kx-ωt) and y =A2in (kx-ωt+ϕ)  resultant wave y = y 1+=2[sin (kx-ωt) + sin (kx-ωt+ϕ)] • finally-> y = 2Acos(ϕ/2) sin (kx-ωt+(ϕ/2)) • resultant wave has the same frequency, wavelength • amplitude of resultant wave is 2Acos(ϕ/2) and if the phase constant =0 then the resultant wave has an amplitude of 2A  waves interfere constructively when cos(ϕ/2)= ±1 and when ϕ = 0, 2 π, 4 π, 6π (even multiple of π)  waves interfere destructively when cos(ϕ/2)= cos(π/2)=0 where ϕ= odd multiple of π • 180 degrees out of phase (crest of one wave meets trough of another)  When ϕ= not an integer multiple of π then the amplitude is between 0 and 2A(not constructive nor destructive o In general: waves in phase-> wave amplitudes add  Waves out of phase-> wave amplitudes subtract o Interference of sound waves  The distance along any path from speaker to receiver is called the path length, r ∣r2−r1 ∣  Difference in path lengths Δr= • If the difference is equal toΔr=nλ (some integer of λ), 2 waves are in a phase and interfere constructively= maximum sound intensity • If the difference is equal toΔr=nλ/2 (n is odd), 2 waves are 180 degrees out of phase and interfere destructively= no sound intensity • Beats: Interference in Time o Beating: periodic variation in amplitude at a given point due to the superposition of two waves having slightly different frequencies (superposition of 2 waves having slightly different frequencies where there is temporal alternations between constructive and destructive interferences) o Beat frequency: number of amplitude maxima one hears per second equals the difference in frequency between 2 sources  Humans can only hear a maximum
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