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 , the number of antinodes a) increases. b) remains the same. c) decreases. d) will equal the number of nodes.

Short answer

Question: When the tension in the string is increased by a factor of 4, the number of antinodes __________. a) remains the same b) increases c) decreases d) cannot be determined Answer: c) decreases

Step-by-step solution

Review wave equation

In order to help us establish a relationship between the tension and the number of antinodes, we must understand the wave equation. The wave equation for a standing wave in a string is given by: v = sqrt(T/μ) where v is the wave speed, T is the tension in the string, and μ is the linear mass density of the string.

Analyze how tension affects wave speed

We can start by determining how changing the tension affects the wave speed. If the tension is increased by a factor of 4, the new wave speed will be: v' = sqrt(4T/μ) v' = 2 * sqrt(T/μ) v' = 2v This tells us that when the tension is increased by a factor of 4, the wave speed doubles.

Relate wave speed to antinodes

To dtermine the number antinodes, we need to define the relationship of wavelength to the fundamental frequency. The wavelength in a string with fixed ends can be determined by the equation: λ = 2L/n where L is the length of the string, λ is the wavelength, and n is the number of antinodes. Recall that the wave speed can be expressed as: v = λf where f is the frequency of the wave.

Determine the effect of increased wave speed on antinodes

Since the wave speed doubles when the tension is increased, we can express the new wave speed and wavelength as: v' = 2v = λ'f From the above equations, we can relate the old and new wavelengths: λ' = 2λ Substitute the λ and λ' expressions to relate the number of antinodes: 2L/n' = 2 * 2L/n n' = n/2

Choose the correct answer

Now that we have derived the relationship between the number of antinodes before and after the tension increase, we can conclude that the number of antinodes will be halved when the tension is increased by a factor of 4. Therefore, the correct answer is: c) decreases.

Key concepts

Wave Equation

Understanding the wave equation is crucial for grasping other concepts in wave dynamics. It mathematically describes how waves propagate through different mediums. For a standing wave, such as in a stringed musical instrument or a science experiment, the equation is expressed as:

\( v = \sqrt{\frac{T}{\mu}} \)

where \( v \) denotes the wave speed, \( T \) the tension in the string, and \( \mu \) the mass per unit length or linear density of the string. When students wonder how the tension in a string affects the sound or behavior of a wave, this equation shows a direct relationship: increasing the tension results in a faster wave speed. However, in the context of standing waves, while speed relates to how quickly the wave oscillates, it's the physical constraints of the string that determine the number of antinodes.

Wave Speed and Tension

The influence of tension on wave speed cannot be understated when it comes to understanding waves on a string. As seen in the solution to our problem, increasing the tension will double the wave speed. How does this work? Because wave speed in a string is a function of both the tension and the linear mass density, and since density is a constant for a given string, modifying the tension will directly alter the wave speed.

When the tension in a string is increased by a factor of 4, as per the square root relationship provided by the wave equation, the new wave speed is consequently doubled. It's a beautiful demonstration of how physical properties like tension can control the dynamics of a wave, and it's expressed via:

\( v' = 2v \)

Understanding this relationship provides valuable insight into how instruments are tuned, how waves carry energy, and even how materials respond to stress.

Wavelength and Frequency

Diving deeper into the nature of waves, we encounter two pivotal concepts: wavelength and frequency. Wavelength (\( \lambda \) is the distance over which the wave's shape repeats, and it is inversely proportional to the frequency (\( f \) - the number of oscillations a wave undertakes in one second.

For standing waves on a string with fixed ends, wavelength is related to the length (\( L \) of the string and the number of antinodes (\( n \) by the formula:

\( \lambda = \frac{2L}{n} \)

This forms the basis for understanding why certain patterns—like the three antinodes in our exercise—emerge at specific frequencies. The wave speed, wavelength, and frequency are bound together in the wave speed equation:

\( v = \lambda f \)

These interconnections are the reason why changing one variable (like tension, affecting speed) impacts the others, creating the fascinating patterns of nodes and antinodes we observe in standing waves.

Standing Wave Properties

Standing waves are a fundamental phenomenon in physics, providing a visual example of wave interference and resonance. A standing wave occurs when two waves of identical frequency and amplitude traveling in opposite directions interfere with each other, producing points of no displacement called nodes and points of maximum displacement known as antinodes.

The properties of these standing waves depend on the medium's characteristics and the boundary conditions. In the exercise, a string fixed at both ends can only support standing waves that fit an integer number of half-wavelengths between the constraints.

By increasing tension and thus increasing wave speed without altering the string length or its fixed ends, we alter the wavelength but not the fundamental constraints that decide the number of nodes and antinodes. The new wavelength is half the original one, but since the string's length doesn't change, the number of antinodes effectively decreases, showcasing a direct connection to the physics governing the system's behavior.