91Ó°ÊÓ

Standing waves are set up on two strings fixed at each end, as shown in the drawing. The tensions in the strings are the same, and each string has the same mass per unit length. However, one string is longer than the other, (a) Do the waves on the longer string have a larger speed, a smaller speed, or the same speed as those on the shorter string? Justify your answer. (b) Will the longer string vibrate at a higher frequency, a lower frequency, or the same frequency as the shorter string? Provide a reason for your answer. (c) Will the beat frequency produced by the two standing waves increase or decrease if the longer string is increased in length? Why? The two strings in the drawing have the same tension and mass per unit length, but they differ in length by \(0.57 \mathrm{~cm} .\) The waves on the shorter string propagate with a speed of \(41.8 \mathrm{~m} / \mathrm{s},\) and the fundamental frequency of the shorter string is \(225 \mathrm{~Hz}\). Determine the beat frequency produced by the two standing waves.

Step by step solution

01

Analyze Wave Speed on Strings

For a string under tension \( T \) with a mass per unit length \( \mu \), the wave speed \( v \) is given by \( v = \sqrt{\frac{T}{\mu}} \). Since both strings have the same tension and mass per unit length, they will have the same wave speed regardless of their length. Hence, the wave speed on both the longer and shorter strings, including the one given at \( 41.8 \, \mathrm{m/s} \), is the same: \( 41.8 \, \mathrm{m/s} \).

02

Determine Frequency on Longer String

The fundamental frequency \( f \) of a string is given by \( f = \frac{v}{2L} \), where \( L \) is the length of the string. With the same speed \( v \) but a longer length for one of the strings, the fundamental frequency decreases as length increases. Hence, the longer string will vibrate at a lower frequency compared to the shorter one.

03

Calculate Beat Frequency

Beat frequency \( f_{beat} \) is the absolute difference between the frequencies of the two waves: \( f_{beat} = | f_1 - f_2 | \). If the longer string's length is increased, its frequency decreases while the shorter string's frequency remains the same, thus increasing the beat frequency.

04

Use Given Data to Solve for Beat Frequency

From Step 1, the shorter string's speed \( v = 41.8 \, \mathrm{m/s} \) and its fundamental frequency \( f_1 = 225 \, \mathrm{Hz} \). For the longer string, use \( v = \frac{v}{2(L+0.0057)} \) to find its fundamental frequency \( f_2 \). Substitute back into \( f_{beat} = |225 - f_2| \) to find the beat frequency.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Wave Speed

Wave speed in a string is influenced by two main factors: the tension in the string and the mass per unit length of the string. The speed at which waves travel along a string can be described by the formula \( v = \sqrt{\frac{T}{\mu}} \), where \( T \) represents the tension and \( \mu \) is the mass per unit length. In situations where the tension and mass per unit length are identical for two strings, the wave speed remains constant. This constancy occurs irrespective of the lengths of the strings. For instance, if two strings have equal tension and mass per unit length but different lengths, their wave speeds will still be the same. Therefore, in such a scenario, if one string has a wave speed of \( 41.8 \, \text{m/s} \), the other will have this same speed, regardless of its length.

Fundamental Frequency

The fundamental frequency of a vibrating string is its lowest possible frequency, at which the string vibrates in its most basic mode. The frequency is given by the equation \( f = \frac{v}{2L} \), where \( f \) is the fundamental frequency, \( v \) is the wave speed, and \( L \) represents the length of the string. For strings with the same wave speed, the length of the string greatly affects the fundamental frequency. A longer string will have a lower fundamental frequency because the wave must travel further to complete one full cycle. Therefore, if one string is longer than another with the same wave speed, it will vibrate at a lower frequency than the shorter string.

Beat Frequency

Beat frequency is an interesting phenomenon that occurs when two waves of slightly different frequencies interfere with each other. The result is a fluctuating sound intensity, which changes at a rate equal to the absolute difference between the two frequencies. This is expressed as \( f_{\text{beat}} = | f_1 - f_2 | \), where \( f_{\text{beat}} \) is the beat frequency, and \( f_1 \) and \( f_2 \) are the frequencies of the individual waves. In our scenario, if the longer string becomes even longer, its fundamental frequency will decrease due to its increased length. Since the frequency of the shorter string does not change, the beat frequency will increase, because the difference between the two frequencies is growing larger.

Tension in Strings

Tension in strings plays a crucial role in determining wave speed. Higher tension results in greater wave speed. For a string fixed at both ends, wave phenomena such as standing waves are observed when the string vibrates. The tension is responsible for maintaining the consistency of wave speed across strings with the same tension and mass per unit length. Important points about tension in strings include:

• Increased tension leads to increased wave speed, given a constant mass per unit length.
• If the tension is equal in two strings, their wave speeds will be identical, assuming equal mass per unit length.
• Changes in wave speed due to altered tension will affect the frequency of vibrations.

By understanding these factors, we gain insights into how tension affects wave behaviors and characteristics, effectively predicting outcomes in string vibration scenarios.