11 Lab Oscillations of a string
Objective: To observe free and forced oscillations of a string. Free oscillations will be observed on a string fixed at both ends by using a weak driving force which varies with time sinusoidally. The waves which come from the movement of the string will result in either produce waves that combine together to double amplitude or cancel out in which cases they are considered resonance or anti-resonance waves respectively. Description: This lab consists of a mechanical wave driver, elastic cord which is hung over a pulley and a weight. The elastic cord is connected to one end of the mechanical wave driver which is flat on the lab table. The other end is pulled and draped over a pulley so a weight can be added to it. We will be using several weights for the experiment and we will need to find out the weight and length of the rope for calculations later in the procedure.
Theory:
Free Oscillations: The transverse displacement of a string fixed with its two ends defined as x = 0 and x = L is called y = (x, t). This is the following equation that describes the string's movement in the y-direction according to Newton's Second Law. 2y - 2y = 0 x2 T t2 The T is the uniform Tension per unit . This equation resolves to v = T/. The v is described by the speed of the waves the towards positive or negative x-axis. All the sinusodial solutions are measured by the following equation: y = f(x) cost, where is the angular frequency and t is the time. When substituted into the first equation, we have d2f + 2f = 0. dx2 T Using frequency v in Hz rather than w in radians/sec, from w=2v the normal mode frequencies for maxima and minima are vn(max) = n T 2L P vn(min) = n T 4L P
In the experiment, the normal modes will be excited by pushing the side of a vibrating rod lightly against the string. The rod will be applied near one end of the string and the coupling is due to friction between the rod and string. Due to damping of the string, which has no been taken into account in the theory, there will be a response of the string for frequencies slightly different from the normal mode frequencies, but the response will fall off in a typical resonance curve fashion as the applied frequencies moves further from the resonant frequencies. The forced oscillations theory predicts a continuous change in the string's response as the frequency is changed monotonically. Data: 200g=.0066/(.073+1.117)= .00550kg/m Mode (n) Theoretical Measured 1 10.259Hz 11.5Hz 2 20.518Hz 22.5Hz 3 30.777Hz 34Hz 4 41.036Hz 45.5Hz (all frequencies observed were at an amplitude of 4V)
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