Problem 6 - University Physics with Modern Physics

Question

A string is made to oscillate, and a standing wave with three antinodes is created. If the tension in the string is increased by a factor of 4 a) the number of antinodes increases. b) the number of antinodes remains the same. c) the number of antinodes decreases. d) the number of antinodes will equal the number of nodes.

Step-by-Step Solution

Answer

#Answer# a) the number of antinodes increases.

Step 1: Understand the relationship between tension and frequency

The frequency of a standing wave on a string depends on the tension in the string and its linear mass density (mass per unit length). The relationship between frequency (f), tension (T), linear mass density (µ), and the speed of the wave (v) can be expressed as: v = sqrt(T/µ) And the frequency is given by: f = v/λ where λ is the wavelength. So, the frequency is directly proportional to the square root of the tension. That is: f ∝ sqrt(T)

Step 2: Determine the effect of increasing the tension by a factor of 4

Since frequency is directly proportional to the square root of the tension, increasing the tension by a factor of 4 will increase the frequency by a factor of: New frequency (f') = sqrt(4) × original frequency (f) f' = 2 × f

Step 3: Understand the relationship between frequency and number of antinodes

In a standing wave, the number of antinodes is directly related to the frequency, and is half the number of harmonics for a closed-closed string (a string with both ends fixed). For instance, in the first harmonic (fundamental frequency), there is one antinode; in the second harmonic (first overtone), there are two antinodes, and so on. So, when the frequency doubles, the number of antinodes increases.

Step 4: Choose the correct answer from the given options

Given the relationships established in steps 2 and 3, when the tension is increased by a factor of 4, the frequency doubles, leading to an increase in the number of antinodes. Therefore, the correct answer is: a) the number of antinodes increases.

Key Concepts

Wave Frequency

Understanding the wave frequency is key when diving into the behavior of standing waves on a medium such as a string. Wave frequency, denoted by the symbol \( f \), is the number of waves passing a point in one second, measured in Hertz (Hz).

In the context of a string oscillating to produce standing waves, frequency is an important indicator of the wave’s characteristics, including its harmonics. The frequency of a standing wave on a string is affected by two main factors: the tension of the string and its linear mass density. As tension increases, frequency also increases because the waves can travel more quickly along the tenser string. This relationship is why tuning instruments such as guitars and violins by adjusting string tension changes the pitch, which correlates directly with frequency.

An intriguing aspect of wave frequency in standing waves is its connection to harmonics. As the frequency changes, the pattern of nodes (points of no displacement) and antinodes (points of maximum displacement) along the string changes too, which is the basis for the creation of different musical notes and tones.

String Tension Physics

The physics of string tension plays a pivotal role in understanding how standing waves form and behave. String tension refers to the force applied along the length of the string, making it taut. This tension is a fundamental factor in calculating the speed \( v \) of a wave traveling through the string, which is governed by the equation:

\[ v = \sqrt{\frac{T}{\mu}} \]
Here, \( T \) represents the tension in the string and \( \mu \) is the linear mass density of the string. As per this relationship, increasing the tension of the string leads to a higher velocity for the wave. Since wave frequency is significantly linked to the speed of the wave, an increase in tension results in an increase in frequency.

It’s crucial to understand that string tension doesn’t just affect the pitch produced by the string but also influences the standing wave’s stability and the production of different harmonic frequencies. High tension can allow for more intricate wave patterns and a wider range of harmonics, highlighting the strong interdependence between the physical properties of a string and the sound waves it produces.

Harmonics in Standing Waves

Delving into the concept of harmonics in standing waves reveals the fascinating ways in which musical instruments create varied sounds. Harmonics are integer multiples of the fundamental frequency—also referred to as the first harmonic or the fundamental. With each consecutive harmonic, the frequency becomes higher, leading to a higher pitch.

Understanding Harmonics

The first harmonic is the simplest standing wave pattern, with one antinode and two nodes at the fixed ends. Each higher harmonic introduces additional nodes and antinodes. For example, the second harmonic would have two antinodes and three nodes, including the fixed ends. The number of antinodes is thus indicative of the harmonic being produced.

In the context of the original problem, where the tension of the string is quadrupled, the frequency is doubled and thus, pushing the standing wave up by one harmonic. This creates more antinodes along the length of the string, altering the resulting sound or note. Therefore, the interplay between tension, frequency, and harmonics is central to the beautiful and complex production of music from a simple string.