Question

Two waves traveling in opposite directions along a string fixed at both ends create a standing wave described by \(y(x, t)=1.00 \cdot 10^{-2} \sin (25 x) \cos (1200 t) .\) The string has a linear mass density of \(0.01 \mathrm{~kg} / \mathrm{m},\) and the tension in the string is supplied by a mass hanging from one end. If the string vibrates in its third harmonic, calculate (a) the length of the string, (b) the velocity of the waves, and (c) the mass of the hanging mass.

Short answer

Answer: The length of the string is \(L = 0.377\,\mathrm{m}\), the velocity of the waves is \(v = 48\,\mathrm{m/s}\), and the mass of the hanging mass is \(m = 11.83\,\mathrm{kg}\).

Step-by-step solution

Calculate the Angular Wave Number (k)

For a given standing wave, the angular wave number (k) can be calculated from the equation: \(y(x, t) = A\sin(kx)\cos(\omega t)\) Comparing this to the given equation: \(y(x, t) = 1.00 \times 10^{-2} \sin (25 x) \cos (1200 t)\) We have \(k = 25 \frac{rad}{m}\).

Calculate the length of the string (L)

In a standing wave vibrating in its nth harmonic, the length of the string (L) can be found using the following formula: \(L = \frac{n\pi}{k}\) Substituting the values for n and k: \(L = \frac{3\pi}{25} = 0.377 \,\mathrm{m}\)

Calculate the Angular Frequency (\(\omega\))

From the given standing wave equation, we can obtain the angular frequency \(\omega\): \(y(x, t) = 1.00 \times 10^{-2} \sin (25 x) \cos (1200 t)\) \(\omega = 1200\,\mathrm{rad/s}\)

Calculate the velocity of the waves (v)

The relationship between the angular wave number (k) and the angular frequency (\(\omega\)) is given by: \(v = \frac{\omega}{k}\) Plugging in the values for \(\omega\) and k, we get: \(v = \frac{1200\,\mathrm{rad/s}}{25\,\mathrm{rad/m}} = 48\,\mathrm{m/s}\)

Calculate the mass of the hanging mass (m)

The tension (T) in the string is the force exerted by the hanging mass (m), which is given by: \(T = mg\) The wave speed v can also be calculated using the formula: \(v = \sqrt{\frac{T}{\mu}}\) Solving for m, we get: \(m = \frac{\mu v^2}{g}\) Substituting the known values: \(m = \frac{0.01\,\mathrm{kg/m} \times (48\,\mathrm{m/s})^2}{9.81\,\mathrm{m/s^2}} = 11.83\,\mathrm{kg}\) Therefore, the solutions to the problem are: (a) The length of the string is \(L = 0.377\,\mathrm{m}\). (b) The velocity of the waves is \(v = 48\,\mathrm{m/s}\). (c) The mass of the hanging mass is \(m = 11.83\,\mathrm{kg}\).

Key concepts

Harmonics

In the world of physics, harmonics refer to the natural frequencies at which a system like a string can vibrate. Imagine plucking a guitar string; it can vibrate in several different patterns or "modes." Each mode corresponds to a different harmonic.
The fundamental frequency is the first harmonic, and it is the lowest frequency at which a string can vibrate. The third harmonic mentioned in the problem indicates that the string vibrates in a pattern where three segments, or "loops," are formed. These loops are sections between nodes, points on the string that remain stationary.
When a string is fixed at both ends, like in the given problem, its entire length decides the harmonic frequencies it can produce. The frequency of the nth harmonic is given by:

• Higher harmonics result in higher frequency vibrations.
• The wavelength becomes shorter as harmonic number increases.
• For the third harmonic on a string, the wavelength is three halves of the fundamental wavelength.

Wave Velocity

Wave velocity is the speed at which waves travel along a medium like a string. It is a crucial parameter in understanding wave behavior. In the context of the problem, wave velocity helps us understand how fast different parts of the string move through their cycles of vibration.
The wave velocity is calculated using the relationship between angular wave number (k) and angular frequency (\(\omega\)). The formula to find wave velocity is:

• \(v = \frac{\omega}{k}\)

Using this formula, you can determine the speed of the wave solely based on these wave properties, which in this case comes out to be \(48\,\mathrm{m/s}\).
This means each wave crest or trough travels along the string at this speed, allowing the wave pattern to continue propagating.

String Vibration

String vibration is an intriguing physical phenomenon that occurs when a string is disturbed, resulting in standing waves. When a string is fixed at both ends, as described, vibrations can produce a pattern of crests and troughs throughout its length. The standing wave equation in this context describes how particular wave patterns form.
The given equation:

• \(y(x, t) = 1.00 \times 10^{-2} \sin(25x) \cos(1200t)\)

is a mathematical way to represent the wave formed on the string. It includes components:\(\sin(kx)\) to account for the spatial vibration profile and \(\cos(\omega t)\) for the time variation part.
Vibrations depend on parameters like tension, mass density, and the wave itself, which determines the overall frequency and amplitude noticed on the string. The resulting behaviors help us study music instruments and various natural waves.

Wave Equation

The wave equation is fundamental to understanding wave behavior. It describes how a wave travels through a medium and establishes the relationship between variables like displacement, velocity, and acceleration of wave packets along the medium.
The wave equation in the context of standing waves involves components like wavelength and frequency. For our example:

• \(y(x, t) = A \sin(kx) \cos(\omega t)\)

Here, \(A\) stands for amplitude, which represents the maximum displacement from equilibrium. \(k\) is the angular wave number, which relates to how many wavelengths fit into a specific length, and \(\omega\) is the angular frequency.
These terms integrate into a formula that describes how every point along the string oscillates, providing a deeper insight into harmonic motion in physics.