Standing Waves Lab Online

Standing Waves Lab Online Purpose The purpose of this activity is to examine the relationships between the tension in a sting, the length density of the string, the length of the string, and the standing waves that can form on the string. Theory A standing wave is a wave where the overall pattern does not appear to move. For standing wave to form on a string, the basic condition that must be met is that both ends of the string must be fixed in place, never moving themselves. This means that for a string stretched out horizontally, the vertical displacement of each of the two ends of the string must always be zero! In a standing wave, the locations of the string that never move are called nodes, while the locations on the string where the vertical displacement will reach its greatest value are called antinodes. Using the condition that there must be a node located at each end of the string for a standing wave to form on it, let us construct a relationship between the length of the string and the size of the waves that can form standing waves on that string. Let the length of the string be 𝐿, then the largest wavelength that will allow for our condition to be met is a wave that is twice as long as the string itself, 𝜆 = 2𝐿 which will result in a standing wave with 1 loop to form on the sting. The speed of a wave 𝑣 is given by the product of its wavelength 𝜆, and its frequency 𝑓. 𝑣 = 𝜆𝑓 Inserting 2𝐿 in for the wavelength, and then solving for the frequency gives. 𝑓1 = 𝑣 2𝐿 1 This equation gives us the fundamental frequency, or the first frequency that will cause a standing wave to form on the string. This frequency is also called the first harmonic. Now let us repeat this process to find the second largest wavelength that will form a standing wave on the string of length 𝐿. The second largest wavelength that will meet the condition of nodes being at both ends of the string is one where its wavelength is equal to the length of the string itself 𝜆 = 𝐿 , which will cause a standing wave with 2 loops to form on the string. Inserting this into the equation for the speed of a wave, and solving it for the frequency we obtain: 𝑣 𝑓2 = , 𝐿 which is the second frequency that will form a standing wave on this string, aka the second harmonic, aka the first overtone. Repeating this process one more time to find the third largest wave that will form a standing wave on the string of length 𝐿. The wavelength of this wave will be related to the length of the string by 𝜆 = which will cause a standing wave with 3 loops to form on the string. Inserting this into the equation for the speed of a wave, and solving for the frequency gives, 𝑓3 = 2𝐿 , 3 3𝑣 , 2𝐿 this, of course, being the third frequency that will form a standing wave on this string, aka the third harmonic, aka the second overtone. From the equations that give the first three frequencies, one should be able to see a pattern for the equations themselves. The second equation is just 2 times the first equation, and the third equation is just 3 times the first equation. 𝑓2 = 𝑣 2𝑣 𝑣 = = 2 ∙ ( ) = 2 ∙ 𝑓1 𝐿 2𝐿 2𝐿 2 𝑓3 = 3𝑣 𝑣 = 3 ∙ ( ) = 3 ∙ 𝑓1 2𝐿 2𝐿 All of the frequencies that will form standing waves on the string follow this basic pattern, such that the 𝑛𝑡ℎ frequency that will form the 𝑛𝑡ℎ standing wave pattern on the string is just 𝑛 times the first frequency 𝑓1 , where 𝑛 is any counting number. 𝑓𝑛 = 𝑛 ∙ 𝑓1 , 𝑛 = 1, 2, 3 … 𝑓𝑛 = 𝑛𝑣 , 𝑛 = 1, 2, 3 … 2𝐿 𝑇 Since we know that the speed of a wave on a string is given by = √𝜇 , where 𝑇 is the tension in the 𝑚 string, and 𝜇 is the length density of the string given by 𝜇 = . We can insert this into our equation to 𝐿 finally obtain 𝑓𝑛 = 𝑛 𝑇 √ , 𝑛 = 1, 2, 3 … 2𝐿 𝜇 Setup 1. Go to the following website: 2. You should now see the following: 3 Procedure: Part 1 1. Near the bottom of the page make the following settings. 𝑘𝑔 a. Linear Density, µ = 5 ∙ 10−3 ⁄𝑚 b. Tension, T = 100 N c. Note that the length of the string is L = 4.0 M d. Record these values in the Table for String Part 1. 2. Near the bottom left of the page change the vibration frequency till you find the lowest frequency that forms a standing wave on this string. The amplitude of the wave will be maximized for the standing wave. Record that value in Table 1. a. Record the number of anti-nodes that this standing wave has in table 1. 3. Find the next 4 frequencies that form standing waves on this string and record them in Table 1 from lowest frequency to highest. Procedure: Part 2 1. At the bottom of the simulation, set the frequency slider to 𝑓 = 60.0 𝐻𝑧 , and the Linear density slider to 𝜇 = 3 × 10−3 𝑘𝑔/𝑚. You can use the arrow keys on your keyboard to make small adjustments if needed. 2. Adjust the Tension slider until a standing wave of n = 8 is formed (this will be the lowest tension you record). 3. Once a standing wave has formed on the string, record the Tension in table 2. 𝑇 4. Calculate the speed of the standing wave using the equation 𝑣 = √ , then record that as the 𝜇 theoretical speed 𝑣 𝑇 in the tables provided. 5. Calculate the experimental wave speed by using the equation 𝑣 = 𝜆𝑓 , then record the experimental speed 𝑣𝐸 in the tables provided. • Wavelength will be the total length (L) divided by the number of loops (n/2). 6. Calculate the % error between the theoretical and experimental values of the speed of the standing wave. 7. Repeat Step 4 through 6 for n = 7, 6, 5, and 4, and record the Tension for each case in the provided tables. 4 Analysis of Standing Waves Lab Name Stefany Pascua Course/Section PHY-1951-005 Physics for Scientist and Engineers 1 Laboratory Instructor Stephen Flowers Table for String Part 1 (5 points) 𝑳(m) 4.0 𝒌𝒈 5*10-3 𝝁( ) 𝒎 𝑻(𝑵) 100 Table Part 1 (10 points) 𝒏 3 4 5 6 7 𝒇𝒏 (𝑯𝒛) 53.25 70.58 88.73 106.05 123.38 Table Part 2 (10 points) 𝒏 8 𝑻(𝑵) 10.45 𝒎 59.02 𝒗𝑻 ( ) 𝒔 𝒎 60 𝒗𝑬 ( ) 𝒔 %error 1.66% 1. For part 1, using 𝑓𝑛 = 𝑛 2𝐿 Table for String part 2 (5 points) 𝒇(𝑯𝒛) 60 𝒌𝒈 3 * 10-3 𝝁( ) 𝒎 𝑳(𝒎) 4.0 7 14.05 68.43 6 19.00 79.58 5 27.55 95.83 4 43.3 120.14 68.57 80 96 120 0.20% 0.53% 0.18% 0.12% 𝑇 √𝜇 calculate the theoretical fundamental frequency 𝑛 = 1 for the standing wave on this string. Show work for credit. (10 points) fn = n(2L)sqrt(T/µ) = 1(2(4.0))sqrt(100/(5*10-3)) = 17.68 Hz 2. Using your data from part 1 and the formula 𝑓𝑛 = 𝑛𝑓1 calculate your average 5 experimental value of the fundamental frequency 𝑓1 of this string. Show work for credit. (10 points) f1 = fn/n = 53.25/3 = 17.75 f1 = fn/n = 70.58/4 = 17.65 f1 = fn/n = 88.73/5 = 17.75 f1 = fn/n = 106.05/6 = 17.68 f1 = fn/n = 123.38/7 = 17.63 f1AV = (17.75 + 17.65 + 17.75 + 17.68 + 17.63)/5 = 17.70 Hz 3. Calculate the % error between the theoretical and your average experimental value for the fundamental frequency of this string. (5 points) % error = |f1 – f1AV|/f1 * 100 = |17.68 - 17.70|/17.68 * 100 = 0.11% 4. From the data from table 2, and using Excel or similar program, graph T (tension) vs n, with the trendline, and show the equation on the graph. Describe the shape of the graph. (5 points) The shape of the graph is linear that does not go through the origin, and as the tension decreases, the harmonic number increases. 5. Using algebra, show that the tension can be written as 𝑇 = (4𝜇𝑓 2 𝐿2 )𝑛2 (5 points) 1 fn = n(2L)sqrt(T/µ) = fn2 = n2/4L2 * T/µ 6 T = (µ4L2fn2)/n2 = (4µL2fn2)/n2 6. Again using the data from table 2, use Excel or a similar program to graph T vs (1/n2), with the trendline, and show the equation on the graph. (5 points) a. From the slope of the graph, calculate the length density of the string. (10 points) 4µf2L2 = 7.92 µ = 7.92/4(60)2(4.0)2 = 3.4375 * 10-5 b. Compare % difference between the given value for the length density of the string, and value you obtained from the slope of your graph. (10 points) 98.86% % error = |µ A – µ E|/µ E * 100 = |3.4375 * 10-5 – 3 * 10-3|/3 * 10-3 * 100 = 7. If the length density of the string were doubled while keeping tension the same, how would the frequency change? (5 points) f = 1/2Lsqrt(T/µ) µ 1 = 2µ f1 = 1/2Lsqrt(T/µ 1) = 1/2Lsqrt(T/2µ) = 1/sqrt(2)(1/2Lsqrt(T/µ)) = f/sqrt(2) 8. Write a general statement that describes how the frequency of vibration of a standing wave relates to the number of segments of the standing wave. (10 points) 7 For a wave, the frequency is the ratio of the speed of the wave to the length of the wave: f = v/λn. Compared to the string length L, wavelengths λn = 2L/n; n = 1, 2, 3, 4, … and correspondence frequencies are given by f = nr/2L; n = 1, 2, 3, 4, ... 8