20200518_PSCI_1422_Lab 3_Standing Wave and Velocity of Sound_Lab_Pokhrel Handout(2) (1)

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p. 1 Physical Science 1422 Lab 3: Standing Waves and Velocity of Sound in Air Group 1 Name: Sahiry Alanis, Karla Duran, Eduardo Chavez, Jocelyn-Ambriz Hernandez The purpose of this laboratory activity is to study concepts of waves, standing waves and the speed of sound in air. The main concepts in this lab are wave properties of sound, resonance, and the standing wave. Vibrating Strings Introduction: The general appearance of waves can be shown by means of standing waves in a string. This type of wave is very important because most of the vibrations of extended bodies, such as the prongs of a tuning fork or the strings of a piano, are standing waves. The purpose of this experiment is to examine how the tension required to produce a standing wave in a vibrating string of fixed length and mass density is affected by the wavelength and the frequency of the wave. Theory: Standing waves (stationary waves) are produced by the addition of two or more traveling waves, which have the same wavelength and speed, but travel in opposite directions through the same medium. Figure 1 shows such a system, where a mechanical vibrator produces a wave on a string which moves to the right and reflection from a fixed end produces a left moving wave. Where the two waves are always 180 0 out of phase, very little motion occurs (none if the amplitudes are the same). Such places are called nodes (see Figure 1). Where the two waves are in phase, the motion is maximum. These positions are call anti-nodes. Figure 2 shows a representation of a standing wave. This representation shows the two extreme positions of the string. This matches well with what we see since the string speed is minimum at the extremes. However, don’t forget that the string goes through all of the positions in between the extremes as shown by the blur in Figure 1. Note that as shown in Figure 2, the node-to-node distance is one-half of the wavelength. The necessary conditions for the production of standing waves on a stretched string fixed at both ends is the length of the string be equal to an integer number of half wavelengths so that there can be a node at each fixed end. Anti-Node Node Node Figure 1 - Standing wave segment on a string

p. 2 Theory: Standing Waves in Strings A stretched string has many natural modes of vibration (three examples are shown in Figure 3 above). If the string is fixed at both ends, then there must be a node at each end. It may vibrate as a single segment, in which case the length ( L ) of the string is equal to 1/2 the wavelength (λ) of the wave. It may also vibrate in two segments with a node at each end and one node in the middle; then the wavelength is equal to the length of the string. It may also vibrate with a larger integer number of segments. In every case, the length of the string equals some integer number of half wavelengths. If you drive a stretched string at an arbitrary frequency, you will probably not see any mode; many modes will be mixed together. But, if the tension and the string's length are correctly adjusted to the frequency of the driving vibrator, one vibrational mode will occur at a much greater amplitude than the other modes. For any wave with wavelength λ and frequency f, the speed, v, is v = λf Eq. (1) The speed of a wave on a string is also related to the tension in the string, F, and the linear density (=mass/length), m, by v 2 = T/m = λ 2 f 2 (Eq. 3) L is the length of the string and n is the number of segments. (Note that n is not the number of nodes). Since a segment is 1/2 wavelength then λ = 2L/n, where n = 1,2,3,… Eq. (3) Solving Equation 2 for the tension yields: T = mλ 2 f 2 Eq. (4) Linear Density (µ) represents an object’s mass per unit length. We can measure linear density using the following formula: Figure 3 - Standing wave representation Figure 2 - Modes of vibration

p. 3 Mass/Length = Linear Density (µ) Eq. (5) In this experiment, we will be finding the tension required to produce a 1 standing wavelength (2 segments) in a string of fixed length and examining the change in required tension as the frequency of the wave is varied. Activity 1: Lab Activity: Standing Wave Vibrating Strings - Lab Report Setup: Use the following website for this part of the lab: https://ophysics.com/w8.html Constant Wavelength () Procedure: In the first part of the experiment, we will hold  constant by always choosing a two-segment pattern so that  = L since n = 2 and vary the frequency while measuring the tension at which a two-segment standing wave appears. By plotting F versus f 2 , we should see a straight line with a slope of m  2 . 1. Use the following initial settings:  Linear Density = 0.3 x 10 -3 kg/m  Frequency = 60 Hz  Tension = 16 N 2. You should see 2 oscillating wave segments on your simulation, as shown below. Remember that 2 segments are equivalent to one complete wavelength! The amplitude (height) of the wave is pretty low and will be maximized at the correct Tension value. Figure 1 - Standing waves simulation with initial values 3. Decrease the tension by clicking the slider under “Tension” once and pressing or holding the DOWN arrow on your keyboard. You can also drag the slider slightly to the left. Continue until your frequency is around 15 N. This should decrease the amplitude, which means we are moving away from our desired Tension value to produce a standing wave.

p. 4 Figure 2 - Slider control for simulation. Click and drag to make rough modifications. Hold up/down arrows on keyboard for smaller adjustments 4. Increase your Tension back to its initial value of 16 N and now increase the tension by clicking the slider underneath Tension once and pressing or holding the UP arrow on your keyboard. At the optimal Tension value, the wave amplitude should be much larger , almost reaching the line above. Input this Tension value in Table 1 below . Figure 3 - Appearance of the standing wave at the optimal Tension value 5. While maintaining linear density and wavelength as constants, use the values for frequency in the Table 1 below, and fill the values for frequency squared ( f 2 ) and the String Tension where the amplitude of the wave is largest. Table 1 - Variable Frequency Linear Density, m (kg/m) Wavelength, λ (m) Frequency, f (Hz) f 2 String Tension, F (N) 0.3 x 10 -3 4 60 3600 17.28 0.3 x 10 -3 4 79 6241 29.96 0.3 x 10 -3 4 100 10000 48.00 0.3 x 10 -3 4 120 14400 69.12 0.3 x 10 -3 4 144.3 20822 99.95 Constant Wavelength Analysis: Graph the values of F vs. f 2 . The plot should be linear. Find the slope of the line that best fits the data. *Note: You can use PASCO Capstone, Excel or other spreadsheet software that you are comfortable using to complete this task . One easy option is to use the following website: https://www.desmos.com/calculator You can simply follow the steps below to graph and export your data: 1. Highlight the f 2 and String Tension columns of your table. Ctrl+C to copy. 2. Go to https://www.desmos.com/calculator 3. Ctrl+V to paste your data into the box. Your plotted points should automatically appear!

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