Explore Suppose a string has a mass of 0.04 kg/m and a length of 0.8 m with fixe

Question

Explore
Suppose a string has a mass of 0.04 kg/m and a length of 0.8 m with fixed endpoints. Find the tension in the string if the fourth harmonic vibrates at 786 Hz

Conceptualize
A wave and its inverted reflection from the fixed end points superpose to form standing waves as illustrated in Active Figure 18.10.

Categorize
We can solve the problem by using the relations between the wavelength and the locations of the nodes from superposing waves of equal wavelength traveling in opposite directions.

Analyze
Following the textbook, in Equation (18.6), it is seen that the nth standing wave, vibrating at n times the fundamental frequency, satisfies:


So the tension is:






T = ?(fn)2 (







2L



n


)2

This can then be evaluated as:






T = (0.04 kg/m)(786 Hz)2 (







2(0.8 m)



4


)2 = N



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Finalize Note how the required tension decreases as the order n of the harmonic that has the required frequency increases. Why does that occur? Note also how the required tension changes with increasing ? or with increasing fn and what leads to that dependence.

Explanation / Answer

on a string =2L/n

f= v/ = n/2L (T/ (M/L))

f= 4/(2*0.8) *sqrt(T/0.04) = 786

T=3954 N

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