These diagrams represent roughly the experimental setup. The first system
was constructed to determine the relationship between the stretched length of
the spring and the tension on the spring. This relationship was used to
identify the tension on the spring in the second system above. The
stretched length of the spring from the supporting rod to the driver was
directly proportional to the tension on the spring. Tension is a useful
parameter for predicting wave velocity.
These diagrams represent roughly the experimental setup. The first system was constructed to determine the relationship between the stretched length of the spring and the tension on the spring. This relationship was used to identify the tension on the spring in the second system above. The stretched length of the spring from the supporting rod to the driver was directly proportional to the tension on the spring. Tension is a useful parameter for predicting wave velocity.
Standing waves were excited on the spring in the second system using a mechanical driver. The frequency of the driving oscillations was recorded (from digital display) and the separation between nodes was measured with a set of calipers. Separations were averaged to calculate wavelength. Velocity was determined from these parameters. The relationship between the wave velocity and frequency was examined to discover whether the spring was dispersive.
Consider a standing longitudinal wave on a spring. Y is the compression on the spring at a position x and time t.
Y(x,t) = A cos (kx-wt) – A cos (kx+wt)
Which simplifies to the following:
Y(x,t) = 2A cos (wt) cos (kx)
Where k = angular wave number (2pi/wavelength) and w = angular frequency (2pi*frequency).
How does the experimental relationship between tension and velocity match up with the theoretical tension velocity relationship for standing waves on a string? Is this model too simplistic, given that mass per unit length is dependent on the tension in the system, and given that the spring itself has significant mass?
V= sqrt(T/mass per unit length) (Serway.)
V= sqrt(T/mass per unit length) (Serway.)
How does the mass of the spring itself affect this system? Nodes were generally closer together spatially toward the bottom of the hanging spring – the mass hanging from the top of the spring, and consequently the tension, was greater than the mass hanging from the bottom of the spring. The mass of the spring was measured to be 14.8 grams. Hanging mass added to the spring varied from 5 grams to about 40 grams, so the mass of the spring is highly significant in this system. Tension at the base of the spring, even with 40 grams hanging mass, is some 30% greater than the tension at the top of the spring.
Data from this investigation shows a highly linear relationship between velocity and tension due to the hanging mass.
Since node separations commonly varied by as much as 20%, and sometimes even 30% in cases with the least external tension, separations were averaged to determine a wavelength to correspond to resonance frequencies. It may have been advisable to add half the mass of the spring to the hanging mass to determine overall tension on the system for the frequency and average wavelength. The best way to deal with this system accurately would probably be to integrate over the spring and hanging mass to determine the tension at a given point on the spring.
According to Serway (some notation changed), the velocity of a wave on a string (fixed at both ends) varies with tension as:
V= sqrt(T/mass per unit length)
This form only holds where the medium is not dispersive – that is where the velocity does not vary with frequency of oscillation.
Is the spring a dispersive medium? According to the data over the frequency and tension ranges examined, the spring is not dispersive. All plots of w vs k are highly linear, showing that the velocity (slope of the graph) does not change with frequency. In this regime, the spring is nondispersive. How closely does this simplistic theory predict observed results?
V=sqrt[T/(.0148/length of string)]
|Spring Length (m)||Predicted V (m/s)||Measured V (m/s)|
These values are close, but consistently greater than the theory predicts that they should be. This error may stem from consistent underestimates of tension due to the hanging mass of the spring itself, as discussed above. Or, this error may simply be do to the not exactly string-y nature of the system.
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