How do I answer questions 1-4? I have typed my data just in case you cannot read my handwriting.

Data Sheet: Standing Waves mass per unit length (gauge) of the string (kg/m) 0 . 018 Kg/m T Trial # hanging Number of string velocity velocity percent mass frequency length half-waves wavelength tension from Eq. 1 from Eq. 2 error (g Hz (m) (m) (N (m/s) (m/s) 1 350 g 20 +2 1:28m 2. 56m) 3.43N 3. 54 m/| 51. 2m/s|- 87.06). 2 250g 30 12 1. 52m 2 1.52 m /2. 45N 3: 85m/s/ 45. Com/5/ - 84.42/1. 3 1450g COD42 1-485 3 1. 123 m / 4, 41 N /4.0com / S/7. 38m/s - 88.6$%. m 4 1450g 70 #2 1. 65m 4 0.815 m / 4. 4IN/3. 99m/s /57. 05m/-86. 92)1. 5 1350 9 82+2 /1: 60m 5 10.650m/ 3.43N/1.01m/s 153.792 ml-06. 125%. 6 250 9 8 8 H2 1. 574m 4 10.50246. /2.45 #3. 93 m/s / 46. 16 m/s / 89.30*- 7 1559 922 + 12 /1.62 m/ 13 10. 249 m) /0.539 4/. 04m/$/22, 908m/s|-70,0%. 8 160 g 97 H2 1.413 12 10.26088m/0.58813. 98mals|26.07mls/-73.$1%. m 9 160g 100H2 1. 594 10. 3542m /1056ENBe 95m/ / 35. 40m/s-79.93%. m 10 1210 9 1.05#12 11595 17 0.4557m /2. 058N/3. com/s /47. 84 m/s |- 84. 712. m 712 (1.574) 250/1000= 1.574 2=0.5 0.25 6 = 0.158 250g 88Hz 6 0.524*88=46.16 -84.30 m 246m 0.25*9.8= V=3.93 m/s 2.45N m/s 2(1.62) 55/1000= 7 1.62 2 =0.24 55g 92Hz 0.055 12 = 0.033 0.249*92=22.90 m 13 9m 0.055*9.8- V=4.04 8m/s -70.0% 0.539N m/'s 60/1000= 8 60g 97Hz 1.613 2(1.613) 12 =0.2 0.06 = 0.037 0.2688*97=26.0 m 688m 0.06+9.8= V=3.98 -73.51% 7m/s 0.58BN m/'s 1.594 2(1.594) 160/10 00= 9 -=0.3 160g 100Hz 9 0.16 = 0.100 0.3542*100=35. m 542m 0.16*9.8= V-3.95 42m/s -79.93% 1.568N m/s 2(1.595) 210/1000= 10 1.595 =0.4 210g 105Hz 7 0.21 7 = 0.131 0.4557*105=47. -84.71% m 557m 0.21*9.8= V=3.96 84m/s 2.058N m/'s Mass per unit length (gauge) of the string (kg/m): 0.018kg/m Questions:Questions 1. Show that Eq. 1 and Eq. 2 are dimensionally reasonable. 2. Explain the relationship between the wavelength 2, and the distance L, from the top of the pulley to the vibrating post. 3. u, the string's mass per unit length, is a 1-dimensional density. Give an example of a 2-dimensional density. What must be the MKS units for a 2-dimensional density? 4. If, at a certain tension, frequency and string length, you produce a standing wave with 6 antinodes, by what factor must you change the tension, leaving the frequency and string length unchanged, to produce a standing wave with 3 antinodes?12. Standing Waves on a Vibrating String The speed of a wave on a string is solely a function of the string's tension and its mass per unit length. At just the right combination of wave speed and string length, the superposition of right and left moving waves on a string will result in a "standing wave", consisting of nodes and antinodes of vibration. Experimental Apparatus In this lab you will be observing standing waves on a horizontal string. One end of the string will be made to vibrate by a computer generated signal whose frequency you will set. Tension in the string will be maintained by attaching weights to its other end, which will be draped over a small pulley. Physical Principles There are many different types of systems in nature which support wavelike motion of one sort or another. Liquids support large amplitude circular waves, in which the liquid molecules move in large circles or ellipses. Solids, liquids and gases all act as media for small amplitude, longitudinal compression waves (sound). Electromagnetic waves consist of a pair of self- inducing electric and magnetic fields which require no medium at all. Unlike sound waves, or for that matter, matter waves of any other kind, they can propagate through a vacuum. Waves on a string - well, they require a string! Waves of various kinds have much in common. They are all disturbances, or "excitations," of their respective media. The details of their mechanisms of propagation differ greatly, but the bottom line is always the same: an elastic property of the medium "communicates" information of the disturbance in one region to an adjacent region. In the case of sound waves this elastic property is the inverse of the medium's compressibility, also known as its "bulk modulus": in the case of waves on a string it is the tension along the string. The greater this elastic property, the faster the wave propagates. On the other hand, since all waves besides electromagnetic waves involve the acceleration of matter, their speed varies inversely with an inertial property of the medium. The greater this inertial property, the slower the wave propagates. In the case of a string the inertial property is the "mass density," or mass per unit length, of the string. Recall that lower notes on a piano, or a guitar, or a violin, are played on looser and fatter strings. A careful dynamical analysis leads to the following formula for the speed of traveling waves on a string: V = VI/M Eq. 1 A purely kinematical analysis (i.e., no forces involved) leads to another expression for the speed of a wave. This approach involves the concepts of 'frequency" (f) and "wavelength" (2). "Frequency" refers to the rate at which crests of the wave pass a given point. Naturally, that is the same as the rate at which troughs pass the point. For best results in this lab choose frequencies 67between 50 and 200 cycles per second. "Wavelength" refers to the distance between crests, or equivalently, between troughs, along a given wave. The speed of the wave is simply the product of these two: v = af Eq. 2 "Standing waves" are waves which appear to be oscillating in place, that is moving neither to the left or to the right. Actually though, they are superpositions of left and right moving waves of equal amplitude. If you like music then this is the lab for you, because standing waves are physically what music is all about, be they standing waves in a pipe (wind instruments), standing waves on a string or a vocal chord, or standing waves on a membrane (percussion instruments). The standing waves, in turn, generate traveling waves in the air (or sometimes, first, in electronic amplifiers), which is what ultimately tickles our ears, but the source of it all is standing waves. In this lab, the wave pulses on the string emanate from the end that is attached to the vibrator. The pulses travel to the other end of the string and rebound off the pulley. When they complete the round trip they rebound again, superposed on subsequent pulses produced by the vibrator. If the timing is just right, that is if the time for a pulse's round trip equals the vibrator's period, or an integer multiple of that period, then the superposed pulses constructively interfere, producing a pulse which is much larger (i.e., has a greater amplitude) than the individual pulses. If the timing is not just right, then the string displays many superposed pulses of random phases and looks like a big mess. The superpositions of one large wave with another large wave moving in the opposite direction is what finally results in the dramatic standing waves that you will see. Where a crest of a right-moving wave meets a trough of a left-moving wave or, a fraction of a period later, a half crest meets a half trough, the string will not be displaced from its equilibrium position. These points are called "nodes." Where crest meets crest or, half a period later, trough meets trough, the string will oscillate with large displacements from its equilibrium position. These points are called "antinodes. " L = n(2/2) relates the number of antinodes, n, the distance between the two fixed ends of the string, L (i.e., from the top of the pulley to the vibrator), and the wavelength 2.Procedure Set up a standing wave on a string. Since you cannot adjust the string's mass density, the frequency, tension, and length will be the "knobs you turn" to tune the system to produce a nice standing wave. The tension is equal to the hanging weight, and therefore can only be varied discretely, although there are some pretty small weights available. The length can be varied continuously by simply sliding the vibrator along the table. The frequency can be varied through the computer. Adjust the tension, the frequency, and the distance between the pulley and the vibrator, to generate 10 different standing waves. Tuning them carefully to generate nice crisp nodes of vibration will pay off in nice low percent errors. Enter the data and the related quantities on the data sheet. Recommended ranges are: hanging mass: 50-250 grams frequency: 20-120 Hertz vibrator to pulley length: 30-180 cm 69Numb Hangi Tri Freque Lengt er of ng Wavelength String Velocity Velocity from Percent al ncy h half Tension (N) from eq mass equation 2 (m/s) error (Hz) (m) wave (m) 1 (m/s) (g) 0.35 350/1000= 1.228 (3.54-51.2 1.28 2(1.28) 2 =2.56 0.35 = 0.273 2.56*20 1 350g 20Hz 1 1 (3.54+51.2 m 0.35*9.8= m V= =51.2m/s 3.43N 100) = 3.43 -87.06% = = 3.54m/ 250/1000= 1.52 2(1.52) =1.52 0.25 = 0.1644 1.52*30=45.6m/ 2 250g 30Hz 2 2 m 0.25*9.8= V=3.85 -84.42% m S 2.45N m/s 450/1000= 1.685 2(1.685) 2 =1.1 0.45 = 0.267 1.123*60=67.38 3 450g 60Hz 3 3 V=4.06 -88.63% m 23m 0.45*9.8= m/s 4.41N m/s 450/1000= 1.63 2(1.63) 2 =0.81 0.45 = 0.276 0.815*70=57.05 4 450g 70Hz 4 0.45*9.8= V=3.99 -86.92% m 5m m/s 4.41N m/s 350/1000= 1.64 2(1.64) =0.65 0.35 = 0.213 0.656*82=53.79 5 350g 82Hz 5 m 0.35*9.8= V=4.01 -86.125% 6m n/s 3.43N m/s