Physics II - PFS&E 20.1-20.2
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This 24 page Class Notes was uploaded by Ishtyaq Ponir on Tuesday September 8, 2015. The Class Notes belongs to at Georgia State University taught by in Summer 2015. Since its upload, it has received 149 views.
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I'm pretty sure these materials are like the Rosetta Stone of note taking. Thanks Ishtyaq!!!
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Date Created: 09/08/15
7 P12 ysjcs 2212K 201 202 I 51 r 39 39 553 Mr39n39 J vgtr39itf31i 39 FUR STIFNTI TIS AND ENGINEFR H 9 h i x A STRATEGIC APPROACH RANDALL DKNIGHT IW Ishtyaq Ponir Dr Jeremy Maune Quick Little Disclaimer Before We Start Hi how is it going I know you39re not typically used to your notes talking to you when you study but here me out before you decide to get right too it As the author I believe that the best way to learn is by means of interaction and communication so I designed these notes to feel like you were studying with a friend We made sure simplify the notes so it reads in ENGLISH not whatever language our professor is speaking I tried my best to make the notes very straight forward and not intimidating How could it be when my font is Happy Monkey seriously no jokes Just relax and together we can get through this and later go about our lives But for now let39s get started Traverse and Longitudinal Waves and Waves in General So the first thing on your mind probably is what the heck a wave is 0 To answer your question a wave is a disturbance traveling through a medium A medium is the area surround a wave while the disturbance is basically a change in what is known as commonplace The medium is almost always stationary not moving as the wave is traveling Alright our first definition a traverse wave is a wave where the displacement how much area the wave is taking from the origin is perpendicular opposite by a right angle from the motion the wave travels 0 Man that was a lot let39s simplify lm If we have a wave ook at the l example we see that the l i direction in which its l l l l l traveling in this case to the 391 1IquotIi39n l3939Irun left IS oppOSIte the pulse the l 3 W My quotcurvequot of the wave in one motion of the wave 0 That39s what we mean when the displacement is opposite to the direction pretty neat huh We also have another type of wave there are two general types with the traverse being the other one called a longitudinal wave 0 This is a case of a wave where the particles in the medium move parallel to the direction of the waves motion Again like before the wave travels to the right but its displacement is influenced by the pushpull of the spring below The traverse wave39s displacement is primarily caused by the updown motion of the string opposite of a longitudinal wave because in this wave it is caused by the leftright pushpull motion of a spring 0 Remember that quotspringquot and quotstringquot are two different words 62 Alright not that we got that out of the way lets focus on a traverse wave for a second In theory it states that the speed not to be confused by velocity in this case is determined the medium 0 This gives rise to a new equation 0 lString lTSIJ N means square root Where Vsmng is the wave speed on the string Ts is the Tension a force opposing the motion and u is known as the linear density The linear density is the strings mass to length ratio That means to get it you have to divide the string39s mass by its lengths as so H mL lt39s is important to note that the wave speed depends on the properties of the medium not the maximum displacement of the wave 0 This means that no matter how much you quotmess withquot the length of those pulses we were talking about oh and by the way multiple pulses on a string at a given instant of position and time is known as a propagation it won39t matter because the speed is determined by the influences outside of the displacement The Fight of the Century Snapshot V History Graphs Before we see Batman and Superman duke it out next year we have to learn the differences of these two very similar but very different graphs First off a snapshot graph is a graph that shows a wave s displacement as a function of position which means it39s a graph literally of a wave and it is what the wave looks like at a single instance of time v uwr w H o This means that the time remains constant as the wave moves about doing what waves do 0 To put it bluntly Iquot Same place in time but position changes 0 A wave pulse moves forward AX that means the change in x relative to time in this case vAt during the time interval At This means that the change in position of the wave is equal to the speed multiplied by the change in time o In the is case the speed is constant Alright now for number 2 a history graph is a graph that shows the wave39s displacement as a function of time at a single position in space 0 Okay things got kinda tricky when we introduced history graphs but bear with me 0 One thing to note is that with a history graph the displacement like the snapshot graph is in the y direction with time now being in the x direction in seconds Alright now let39s compare and contrast the two graphs provided below the left being the snapshot and the right being the history graph 0 We see that both graphs have their displacements D in the y direction o The two also have a certain framework or constant from which the graphs seem to sort of quotstand stillquot by that constraint For the snapshot graph it39s the little time symbol t and for the history graph it39s the little position symbol x o This is important for converting from snapshot to history graphs which we will cover later 0 Now the only real values that are different for the two graphs are the values in the X direction for snapshot graphs its position and for history graphs its time 0 Here39s the fun part in order to understand their differences we have to look at how they relate to one another For one the history graph shows how the position at x meters changes woughtune The snapshot graph shows how the position changes at a single instance of time So you can apply what happens to a snapshot graph to a history graph 0 First see what position the history graph is in this will be important when we look at the snapshot graph 0 Now on the snapshot graph we track the position that was on the history graph or we move the snapshot graph by its wave speed c We repeat this and while we do so we see the value of the displacement of the position we are using on the history graph and plot that displacement on the history graph 0 That39s really all too it and you continually do this and connect the points you plotted To do a history to snapshot graph you39d do the same thing but instead you would see how different times would change given their positions A x That should cover the basics of these two types of graphs make sure to check out the PracticeSet to see how to work some practice problems in the coming week The Sinusoidal Wave The Wave That Moves Like a Circle Sinusoidal waves are wave sources that oscillate with Simple Harmonic Motion SHM o SHM are basically motions that oscillate move in circle pattern that have a force that hinders them which is proportional to the motions displacement Each one of the particles in the medium undergoes SHM with frequency 19 o f 77quot 0 Units 5 1 Hz 0 Frequency is equal to 1 divided by the period 0 This is so because frequency is equal to the number of oscillation per second so 1 divided by the period is equal to the frequency 0 This is true because as the wavelength ends we get the result of the time it takes for that wavelength to finish so taking on over that takes the amount of oscillations it takes for that duration of time o The distance spanned by one cycle of the motion is called the wavelength 1 This is basically the ami itude distance between a I crest the top of a wave to another crest or a trough the bottom of the wave to trough This is the basic measure of length that we have for analyzing waves 0 Amplitude is defined as the maximum displacement of the wave In other words the wave s highest height from the source its girth so to speak It is measured from the base to either the crest of the wave the highest height in the positive y direction or from the base to the trough the lowest height in the y direction As we ve said before or might not have the size of the waves amplitude is i independent of the wave speed so quite literally size does not matter 0 The way we measure speed yes speed not velocity remember were looking for scaer qualities not vectorones is by taking distancetime Fair enough but what are the distance and the time Well we figured out that A is the distance of the wavelength and T is the period of a wave or the time it takes to reach one wavelength So with that logic we say Vspeed lT or the wavelength divided by the pe od Note a wave moves forward one wavelength in one period of time Note remember to not confuse the motion of the wave with the motion of the medium we discussed this earlier 0 So from the equation we gathered earlier like three bullet points ago we can say that V if c This is known as the fundamental rea Honsquot0 of periodic waves 0 This means that a wave moves forward a distance of one wavelength during a time interval of one period c We get 1 from this equation because 1 is equal to 1T or the 1 divided by the period 0 Through some simple rearranging we get the equation you see above see not so hard 0 We can further rearrange and say that A Vf o It is important to note that VlS a property of the medium caused by it and f is a property of the source caused by that 0 So because of this we say the wavelength is a consequence of a wave of the frequency traveling through a medium in which the wave speed is V Now let39s move on to displacement To calculate displacement you use the equa on o DXz 4539n2TXZ by D is of course the displacement taking into account its X position and t time values DalS called the phase constant which is defined by the initial conditions of a wave how the wave started its origins This equation can be altered in different was but still mean the same thing o D ZA539n21tXZ Do A5n21tXZ 45021t rad 0 Note the A s cancel out A5h21TXZ qbg o A5n21tXZ Do pm 0 The x A before the 2T comes from the fact that were taking the displacement from the quantity defined by D Z c The 2 T rad comes from the equation 539na 2n rad 539na o This is a pre calculus proof that states that adding 2T to the a trigonometric equation does not change the value of the equation but its position 0 The same goes for the A added to X the value is the same just the positions change The disturbance at X Z is exactly the same as the disturbance at X Now that we have the equation of the displacement we have to set the quotwave in motionquot We does this by replacing 39X39 with the value quotx Iz39quot o This works because the wave moves a distance of V2 during time 2 0 And like before when we said adding to any expression changes its positon not displacement that idea comes full circle literally 0 We subtract because we are taking an earlier time 2 20 and comparing it tojust X 0 SO 39 0061 DOMl i0 And Soagain o DXz 4539727TX V172 DO DMUASI7727TXZl U o 0 Note we used V22 fz T to write VZ 77quot So now we have the formula lets add more variables to make to more complex 0 Because sinusoidal waves propagate using SHM we can inherit variables used by those equa ons o The first one we are going to look at is angular frequency oo2nf27TT remember how frequency and period relate o Units rads Squot1 0 Next we have the wave number k ZITA o Units radm m 1 We can now simplify the displacement equation using these new quantities o DXz 4539nltXw t DO Note it repeats every t seconds Note if x is fixed then the equation gives a sinusoidal snapshot graph at one point in space t1 otherwise it is a history graph We essentially switched out values to get to this equation nothing to fancy This still accounts for a sinusoidal wave traveling in the x direction this case is positive In the negative X direction its o D 172A5hltXcu z DO 0 So since we described the phase constant being the initial conditions the equation for it would be I D0m05A539ncbo o The X and t cancel out the k and co so we are left with the phase constant 0 Note different values of DO describe different initial conditions for a wave The equation we came up with Dz A5397ltXcu t DO is the displacement in the X direction We can do the same in the y direction o y 27245hX cu l DO o The velocity of a particle which is not the same as the velocity of the wave along the string is found by taking the derivative with respect to time Vy dydz z calCOSfX cu t Do 0 This is accomplished by taking the derivative of the equation 0 First do chain rule so take the derivative of the quotinsidequot multiplied by the derivative of the quotoutsidequot with the quotinsidequot intact c When taking the derivative of the quotinsidequot 0 kx and Do disappear because they are constants we can get rid of them because they are separated with a and if it were multiplication or division it would be a completely different case o The co is the only variable that relates to time so we take its derivative co 0 When taking the derivative of the quotoutsidequot we get A multiplied by the cosine of whatever we had on the inside to start 0 We then multiply the two parts quotoutsidequot and quotinsidequot and get the equation listed above 0 The maximum velocity of the string is VmaXcm4 0 Again we applied the rules of SHM to get this To find the expression for a wave speed along a string we need to look at Newton39s laws specifically his second one 0 Of we use Ax to signify a small portion of length A we can use it in our equation 0 Next we measure the string39s tension as in exerts a downward force on the piece of string which is pulling it back to equilibrium 0 Because of this fact we can use the equation for Newton s Second Law Fnedfmay From this we can substitute m with uAX c We get this because u is an objects mass to length ration so to get mass we multiply the little piece of A A X and it gets rid of the A length on the bottom of the linear density39s mass to length ratio To further our discussion of displacement we can look at the point of maximum displacement to also find the point of maximum acceleration This is also a time derivative so it can be written 0 aydvydz MA5hXw z CDo And because of this we can say the crest of the wave is equal to o ay a24 This is fine and dandy but we have to remember that the angular frequency of which the particles of the string oscillate is related to the wave s speed Ialong the string by the equation wvk 0 SO I ay a 2A 42sz o The v2 and the k2 come from the fact that we are squaring oo So that concludes sinusoidal waves for now I ve made sure to add Waves in Two and Three Dimensions next week just in case you needed some time to read through these if you needed a break from physics or if you just don t want to read my notes anymore Anyways Thank you for taking the opportunity to better yourself and to study with me o I hope you stick by me we have a lot more ground to cover 0 Until next time stay classy
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