Physics Experiments 132 3-Xxx

PHYSICS EXPERIMENTS — 132 3-1

Experiment 3 Vibrating Strings

In this experiment you shake a stretched flexible Standing waves will be produced when the string. When you shake rhythmically at the correct length of the cord is an integral number n of half frequency, the string forms a large standing wave wavelengths, pattern. The vibrating string resonates at these L = n l 2 (eq. 1) frequencies, which are determined by the properties ( n ) of the system. You use your observations to where n is any positive integer and n is the determine the wave speed on the string. corresponding wavelength, as shown in Figure 2 (for n = 4). The wave speed v can be related to the

Preliminaries. frequencies fn at which resonance occurs, This experiment investigates the resonance v = nfn (eq. 2) conditions of a simple system consisting of a taut cord fixed at the ends and connected to an external driving mechanism.

Figure 1. Schematic for Standing Waves Experiment

Waves travel back and forth along the cord when The wave speed v depends only on the linear excited by the driver. A standing wave can be mass density  of the string ( = m/L, m is the mass produced by waves of the same frequency traveling of the string) and the tension F in the string by in opposite directions on the cord. The required T driving frequencies are determined by the properties v = FT /m (eq. 3) of the string; the tension FT, the string mass m and Eq. 2 allows an experimental determination of the string length L. When a resonant frequency is wave speed by observation of standing waves. Eq. used to drive the string, the maximum amplitude of 3 allows an independent calculation of the wave vibration of the string will be much larger than the speed from the measured properties of the string. amplitude of the vibration of the driver and is easily seen. The standing wave pattern is fixed in space, as shown in Figure 2. The points which remain Procedure. stationary are called nodes, while the points which  Select a bungee cord. Measure and record its undergo maximum oscillation are called antinodes. mass.

Figure 2. Example of a Standing Wave Pattern

node antinode

 Hook the bungee cord at one end over the stand and stretch the other end using the 11- 2 PHYSICS EXPERIMENTS — 132

force scale until the tension force read on the scale is about 5-6 N. Measure and record Questions (Answer clearly and completely). the tension and the cord length. 1. Take your resonant frequencies and divide each by the corresponding number of antinodes. Is there  Clip the speaker clip to the cord near one a pattern? What is its significance? end. Adjust the frequency on the driving 2. Does your data form a straight line on the graph audio generator to produce a standing wave. of wavelength vs. frequency? Should it? The amplitude (volume) does not need to be turned up very high. Setting it too high may 3. What value do you determine experimentally for damage the speaker!! At resonance (when the wave speed? What is the percent difference of the frequency is equal to one of the this value from the theoretical value? frequencies that gives standing waves) the amplitude of the waves will be larger than 4. Suppose you redo the experiment with the same for nearby frequencies and the nodes should cord pulled to a longer length. Explain how the be quite distinct. Vary the frequency to find following quantities change: string tension, mass the largest amplitude. Record the resonance density, wave speed, wavelengths of standing frequency fn , the number of antinodes n, and waves, frequencies of standing waves. How would the average distance between nodes. the graph of reciprocal wavelength on the vertical axis and frequency on the horizontal axis change?  Calculate the wavelength from the average distance between nodes. rev. 8/13

 Look for at least four other resonant frequencies. Record the frequency and the wavelength for each standing wave.

 Graph the data with the "wavelength" on the vertical axis and "frequency" on the horizontal axis.

 Make another graph with "wavelength" on the vertical axis and "inverse frequency" on the horizontal axis. Eq. 2 predicts that the data on this graph should lie on a straight line. Calculate the wave speed from the slope of the graph.

 Calculate the wave speed from the properties of the stretched string, using eq 3.

 The experimental calculation of wave speed (from the graph) should agree with the theoretical determination of the wave speed (from eq 3.). Determine the percent difference of the experimental value from the theoretical value.