Question

The largest tension that can be sustained by a stretched string of linear mass density \(\mu\), even in principle, is given by \(\tau=\mu c^{2},\) where \(c\) is the speed of light in vacuum. (This is an enormous value. The breaking tensions of all ordinary materials are about 12 orders of magnitude less than this.) a) What is the speed of a traveling wave on a string under such tension? b) If a \(1.000-\mathrm{m}\) -long guitar string, stretched between anchored ends, were made of this hypothetical material, what frequency would its first harmonic have? c) If that guitar string were plucked at its midpoint and given a displacement of \(2.00 \mathrm{~mm}\) there to produce the fundamental frequency, what would be the maximum speed attained by the midpoint of the string?

Short answer

Based on the step-by-step solution provided: a) The speed of the wave traveling on this string under maximum tension is equal to the speed of light in a vacuum, \(c\). b) The first harmonic frequency of the guitar string is \(\frac{c}{2}\). c) The maximum speed attained by the midpoint of the string is \(\sqrt{\frac{\mu c^2 x^2}{m}}\).

Step-by-step solution

Compute the wave speed on the string under maximum tension

To find the wave speed, we can use the formula for wave speed on a string, which is given by \(v = \sqrt{\frac{T}{\mu}}\), where \(v\) is the wave speed, \(T\) is the tension, and \(\mu\) is the linear mass density. We are given that the maximum tension \(\tau = \mu c^2\), so we can substitute this into the formula to find the wave speed: \(v = \sqrt{\frac{\tau}{\mu}} = \sqrt{\frac{\mu c^2}{\mu}} = \sqrt{c^2} = c\). The speed of the wave traveling on this string under maximum tension is equal to the speed of light in a vacuum, \(c\).

Find the frequency of the first harmonic

For a string that is fixed at both ends, the first harmonic frequency can be found by using the formula: \(f_1 = \frac{v}{2L}\), where \(f_1\) is the first harmonic frequency, \(v\) is the wave speed, and \(L\) is the length of the string. In this case, we have a \(1\) meter long guitar string (\(L = 1.000 \, \text{m}\)) with wave speed \(v = c\). Substitute the values of \(v\) and \(L\) into the formula: \(f_1 = \frac{c}{2(1.000 \, \text{m})} = \frac{c}{2}\). The first harmonic frequency of the guitar string is \(\frac{c}{2}\).

Compute the maximum speed of the string's midpoint

We can use the energy conservation principle to find the maximum speed of the string's midpoint when it is plucked. The maximum potential energy stored in the string when it is plucked will be equal to the maximum kinetic energy when the string midpoint reaches its maximum speed. The maximum potential energy \(U_{max}\) can be calculated as: \(U_{max} = \frac{1}{2} k x^2\), where \(k\) is the spring constant, and \(x\) is the displacement. The maximum kinetic energy \(K_{max}\) can be calculated as: \(K_{max} = \frac{1}{2} m v_{max}^2\), where \(m\) is the mass of the string's midpoint, and \(v_{max}\) is the maximum speed. By equating the maximum potential energy and the maximum kinetic energy, we get: \(\frac{1}{2} k x^2 = \frac{1}{2} m v_{max}^2\). We can find the spring constant \(k\) by using the formula for the tension of the string \(T = kx\). From step 1, we have \(T = \mu c^2\). \(k = \frac{T}{x} = \frac{\mu c^2}{x}\). By substituting the value of \(k\) in the energy equation, we get: \(\frac{1}{2} (\frac{\mu c^2}{x}) x^2 = \frac{1}{2} m v_{max}^2\). Rearranging the equation to solve for \(v_{max}\): \(v_{max} = \sqrt{\frac{\mu c^2 x^2}{m}}\). Finally, the maximum speed attained by the midpoint of the string is \(\sqrt{\frac{\mu c^2 x^2}{m}}\).

Key concepts

Traveling Wave

A traveling wave is characterized by a disturbance that moves through a medium over time. When we pluck a guitar string, we create a pulse that travels back and forth between the fixed ends of the string, thus creating a traveling wave. The energy is transferred through the string without the mass of the string itself moving along the length; instead, the material of the string oscillates around an equilibrium position.

Through this process, the particles of the medium (in this case, the string) undergo periodic motion—first moving upwards from their rest position, then downwards, passing through the rest position again, and continuing this way. This creates crests and troughs, which are characteristic of traveling waves. Where the medium's particles cluster together, we call them compressions, and where they spread out, they form rarefactions.

Harmonic Frequency

The harmonic frequency of a string can be thought of as the 'natural' frequency at which the string vibrates. Each time the string vibrates back and forth once, we call that a cycle, and the number of cycles per second is what we refer to as frequency, measured in Hertz (Hz). The first harmonic, also known as the fundamental frequency, is the lowest frequency at which the string can sustain a standing wave. That is to say, it's the simplest form of vibration for the string, with nodes located only at the fixed endpoints, and an antinode, or point of maximum displacement, at the center.

For a string of length L under tension and fixed at both ends, the first harmonic is derived by determining the wave speed and dividing it by twice the length of the string, following the formula \( f_1 = \frac{v}{2L} \). Higher harmonics, or overtones, occur at frequencies that are integer multiples of the fundamental frequency.

Maximum Tension on a String

The maximum tension a string can sustain is of critical importance to understanding the behavior of a string under stress. In physics, we often assume ideal conditions, such as a perfectly elastic string with uniform mass distribution. Under these ideal conditions, the hypothetical maximum tension that a string can endure, as shown in the given problem, would be its linear mass density \( \mu \) times the square of the speed of light \( c^2 \).

However, in the real world, materials have their breaking points far below this theoretical value, and the maximum tension they can withstand is usually less by many orders of magnitude. It's a theoretical exercise to imagine what would happen if a string could sustain such high tension and the effects it would have on the wave speed, which as we observed, would equal the speed of light.

Linear Mass Density

Linear mass density \( \mu \) is the measure of mass per unit length along a string or other one-dimensional object. It's an intrinsic property of the material that composes the string and is independent of the total mass or length of the string. The linear mass density plays a crucial role in determining the behavior of waves on the string.

In the context of our exercise, the relationship between linear mass density and wave speed is given by the equation for the speed of a wave on a string: \( v = \sqrt{\frac{T}{\mu}} \), where \( T \) is the tension in the string and \( v \) is the wave speed. This illustrates that for a given tension, the lighter the string (lower \( \mu \) ), the faster the wave propagates along it.

Speed of Light

The speed of light in a vacuum, denoted as \( c \) and approximately equal to \(3 \times 10^8 \) meters per second, is a fundamental constant of nature. This speed represents the fastest velocity at which information or energy can travel through the vacuum of space. According to modern physics, nothing can travel faster than the speed of light.

In the hypothetical scenario of our exercise, a string under tension that equals its linear mass density times the square of the speed of light would propagate a wave at the speed of light itself. This concept underscores the relationship between mass, energy, and the fine-structured fabric of spacetime, theorized in Einstein's equations, where the energy tied up in mass is equal to the mass times the speed of light squared \( (E = mc^2) \). The example discussed is an interesting thought exercise in the realm of theoretical physics.