Maximum Displacement Of A Tuning Fork Explained

In the realm of physics and acoustics, understanding the behavior of vibrating objects is crucial. This article delves into the fascinating world of tuning forks, exploring how their displacement can be mathematically modeled and, more importantly, how to determine their maximum displacement. We will dissect the given equation, $d=0.4 ext{sin}(1760πt)$, and use our knowledge of trigonometric functions to arrive at the solution. This exploration will not only provide the answer to the question but also enhance our understanding of oscillatory motion and the underlying principles of wave mechanics. Let's embark on this journey of discovery, unraveling the secrets of tuning fork vibrations and their mathematical representation.

Dissecting the Displacement Equation

The displacement equation provided, $d=0.4 ext{sin}(1760πt)$, is a sinusoidal function that describes the movement of the tuning fork over time. This equation is a cornerstone of understanding the behavior of the tuning fork, and each component within it plays a crucial role. The variable d represents the displacement in millimeters, which is the distance the tuning fork moves away from its resting position. The variable t represents time in seconds, the independent variable that drives the oscillation. The sine function, $ ext{sin}(1760πt)$, is the heart of the oscillatory motion, capturing the rhythmic back-and-forth movement of the tuning fork. The coefficient 0.4 in front of the sine function is particularly important as it dictates the amplitude of the oscillation, which directly relates to the maximum displacement. Understanding these individual components and their interplay is key to deciphering the equation and answering the question at hand. Furthermore, the term $1760π$ within the sine function represents the angular frequency, which determines how fast the tuning fork vibrates. This angular frequency is directly related to the frequency of the sound produced by the tuning fork, a crucial aspect in understanding its acoustic properties. By carefully analyzing each component of the equation, we gain a deeper appreciation for the physics behind the tuning fork's motion and its mathematical representation.

Understanding Amplitude and Maximum Displacement

In the context of sinusoidal functions, amplitude is the maximum displacement of the oscillating object from its equilibrium position. In simpler terms, it's the peak value that the function reaches. For a function of the form $d = A ext{sin}(Bt)$, where A represents the amplitude and B influences the frequency, the amplitude directly corresponds to the maximum displacement. This concept is crucial for understanding the behavior of oscillating systems, including the tuning fork in our problem. The amplitude provides a clear indication of the range of motion of the vibrating object, and it's a key parameter in characterizing its oscillatory behavior. A larger amplitude signifies a greater displacement from the equilibrium, indicating a more energetic oscillation. Conversely, a smaller amplitude implies a smaller displacement and a less energetic oscillation. In the case of the tuning fork, the amplitude directly corresponds to the maximum distance the tines move away from their resting position during vibration. Therefore, understanding the concept of amplitude is essential for determining the maximum displacement of the tuning fork and interpreting the physical meaning of the displacement equation. The amplitude is not just a mathematical parameter; it's a direct reflection of the physical characteristics of the oscillating system.

Determining the Maximum Displacement

To find the maximum displacement of the tuning fork, we need to leverage our understanding of the sine function. The sine function, $ ext{sin}(x)$, oscillates between -1 and 1. This inherent property of the sine function is critical in determining the maximum and minimum values of any sinusoidal function. When we have an equation of the form $d = A ext{sin}(Bt)$, the maximum value of d occurs when $ ext{sin}(Bt)$ reaches its maximum value, which is 1. Conversely, the minimum value of d occurs when $ ext{sin}(Bt)$ reaches its minimum value, which is -1. Therefore, the maximum displacement is simply the absolute value of the coefficient A, which represents the amplitude. In our case, the equation is $d = 0.4 ext{sin}(1760πt)$. The coefficient in front of the sine function is 0.4. This means that the amplitude, and hence the maximum displacement, is 0.4 millimeters. The sine function's oscillatory nature ensures that the displacement fluctuates between -0.4 mm and +0.4 mm, with 0.4 mm being the furthest distance the tuning fork moves from its resting position in either direction. This direct relationship between the coefficient and the maximum displacement allows us to quickly and easily determine the peak value of the oscillation.

Applying the Concept to the Problem

In our specific problem, the equation given is $d=0.4 ext{sin}(1760 ext{π} t)$. By directly comparing this equation to the general form $d = A ext{sin}(Bt)$, we can immediately identify that $A = 0.4$. As we established earlier, the amplitude A represents the maximum displacement. Therefore, the maximum displacement of the tuning fork is 0.4 millimeters. This is the peak distance the tuning fork moves away from its equilibrium position during its vibration. The term $1760πt$ inside the sine function determines the frequency of oscillation, but it does not affect the maximum displacement. The maximum displacement is solely determined by the amplitude, which is the coefficient 0.4 in this case. This straightforward application of the concept allows us to quickly arrive at the correct answer without needing to perform any complex calculations. The maximum displacement is a crucial parameter in understanding the behavior of the tuning fork, as it dictates the range of its motion and the intensity of the sound it produces. This direct connection between the equation and the physical phenomenon highlights the power of mathematical modeling in understanding the world around us. Therefore, the correct answer to the question is B. 0.4 mm.

Conclusion

In conclusion, by understanding the relationship between the displacement equation, the sine function, and the concept of amplitude, we were able to determine that the maximum displacement of the tuning fork is 0.4 millimeters. This problem highlights the importance of understanding the fundamental properties of sinusoidal functions and how they can be used to model real-world phenomena. The amplitude, represented by the coefficient in front of the sine function, directly corresponds to the maximum displacement in oscillatory systems. This principle is not only applicable to tuning forks but also to a wide range of physical systems, including pendulums, springs, and even electrical circuits. The ability to analyze and interpret equations like $d=0.4 ext{sin}(1760πt)$ is a valuable skill in physics and engineering, allowing us to predict and understand the behavior of oscillating systems. Furthermore, this exploration underscores the power of mathematics as a tool for describing and understanding the natural world. By mastering these fundamental concepts, we can gain deeper insights into the workings of the universe and appreciate the elegance of mathematical models in capturing complex physical phenomena.

Maximum Displacement, Tuning Fork, Sinusoidal Function, Amplitude, Oscillation, Displacement Equation