What Is The Maximum Displacement Of The Tuning Fork Given The Equation D = 0.4 Sin(1760t)?
In the realm of physics and acoustics, understanding the behavior of vibrating objects is crucial. Tuning forks, known for their ability to produce pure tones, serve as excellent examples for exploring wave phenomena. This article delves into the mathematical representation of a tuning fork's displacement over time, focusing on determining its maximum displacement. We will dissect the given equation, identify key parameters, and apply trigonometric principles to arrive at a solution. This exploration is not just an academic exercise; it provides insights into the fundamental principles governing sound production and wave mechanics. By understanding the mathematical model, we can better appreciate the physics behind musical instruments and other vibrating systems. So, let's embark on this journey of mathematical discovery and unravel the secrets of the tuning fork's motion.
Deconstructing the Equation: d = 0.4 sin(1760t)
At the heart of our analysis lies the equation d = 0.4 sin(1760t), which models the displacement (d) of the tuning fork in millimeters as a function of time (t) in seconds. This equation is a classic example of a sinusoidal function, specifically a sine wave. Let's break down each component to understand its significance. The sine function, denoted as sin, is a fundamental trigonometric function that oscillates between -1 and 1. This oscillatory nature perfectly captures the back-and-forth motion of a vibrating object like a tuning fork. The coefficient 0.4 in front of the sine function represents the amplitude of the wave. Amplitude is the maximum displacement from the equilibrium position, essentially how far the tuning fork moves in either direction. In this case, the amplitude is 0.4 millimeters. This means the tuning fork vibrates 0.4 mm away from its resting position in both directions. The value 1760 inside the sine function, multiplied by t, is related to the frequency of the vibration. Frequency determines how many cycles of vibration occur per second. A higher frequency corresponds to a higher-pitched sound, while a lower frequency corresponds to a lower-pitched sound. In this equation, 1760 is related to the angular frequency, which we'll discuss further. Understanding these components is crucial for interpreting the equation and extracting meaningful information about the tuning fork's behavior. The amplitude directly tells us the maximum displacement, while the frequency dictates the tone produced. By analyzing the equation, we can connect the mathematical representation to the physical phenomenon of sound generation.
Identifying Key Parameters
To effectively analyze the equation d = 0.4 sin(1760t), we need to pinpoint the key parameters that govern the tuning fork's motion. As mentioned earlier, the coefficient 0.4 is the amplitude. Amplitude, in the context of wave motion, represents the maximum displacement of the oscillating object from its equilibrium position. In simpler terms, it's the farthest the tuning fork moves from its resting point. In this case, the amplitude of 0.4 millimeters signifies that the tuning fork vibrates 0.4 mm away from its center position in both directions. This value directly determines the maximum displacement we are trying to find. The value 1760 inside the sine function plays a crucial role in determining the frequency of the vibration. While 1760 isn't directly the frequency in Hertz (cycles per second), it is related to the angular frequency, denoted by ω (omega). The relationship between angular frequency (ω) and frequency (f) is given by the equation ω = 2πf. In our case, ω = 1760. To find the frequency in Hertz, we would divide 1760 by 2π. The frequency determines the pitch of the sound produced by the tuning fork. A higher frequency corresponds to a higher pitch, and a lower frequency corresponds to a lower pitch. The time variable t represents the elapsed time in seconds. As time changes, the value of sin(1760t) oscillates between -1 and 1, causing the displacement d to change accordingly. Understanding the role of each parameter – amplitude, angular frequency, and time – is crucial for interpreting the equation and predicting the tuning fork's motion. The amplitude directly tells us the maximum displacement, while the angular frequency dictates the rate of oscillation and the resulting sound pitch. By carefully examining these parameters, we can gain a deeper understanding of the tuning fork's behavior.
Determining Maximum Displacement: The Role of the Sine Function
The key to finding the maximum displacement lies in understanding the behavior of the sine function, sin(1760t). The sine function is a periodic function that oscillates between -1 and 1. This means that the value of sin(1760t) will never be greater than 1 or less than -1, regardless of the value of t. The maximum value of the sine function is 1, and the minimum value is -1. These extreme values are crucial for determining the maximum displacement of the tuning fork. In the equation d = 0.4 sin(1760t), the displacement d is directly proportional to the sine function. When sin(1760t) reaches its maximum value of 1, the displacement d will also reach its maximum value. Conversely, when sin(1760t) reaches its minimum value of -1, the displacement d will reach its minimum value (which is the maximum displacement in the opposite direction). To find the maximum displacement, we simply need to consider the case where sin(1760t) = 1. Substituting this value into the equation, we get: d = 0.4 * 1 = 0.4. Therefore, the maximum displacement of the tuning fork is 0.4 millimeters. This result highlights the importance of the sine function's range in determining the maximum and minimum values of the displacement. Since the sine function oscillates between -1 and 1, it effectively scales the amplitude (0.4 mm in this case) to determine the extremes of the tuning fork's motion. By understanding this relationship, we can easily find the maximum displacement for any sinusoidal motion described by a similar equation. The maximum displacement corresponds to the amplitude of the sinusoidal function.
Applying Trigonometric Principles
To solidify our understanding of the maximum displacement, let's delve deeper into the trigonometric principles at play. The sine function, a cornerstone of trigonometry, graphically represents the y-coordinate of a point moving along the unit circle as a function of the angle. As the angle increases, the point moves around the circle, and the y-coordinate oscillates between -1 and 1. This oscillation is precisely what gives the sine function its characteristic wave-like shape. The equation d = 0.4 sin(1760t) leverages this oscillatory behavior to model the tuning fork's motion. The argument of the sine function, 1760t, represents the angle in radians. As time (t) increases, the angle changes, causing the sine function to oscillate. The amplitude, 0.4, scales the output of the sine function, determining the maximum displacement. The maximum value of sin(1760t) occurs when the angle 1760t is equal to π/2 radians (90 degrees) plus any integer multiple of 2π radians. At these angles, the y-coordinate of the point on the unit circle is 1, resulting in the maximum value of the sine function. Conversely, the minimum value of sin(1760t) occurs when the angle 1760t is equal to 3π/2 radians (270 degrees) plus any integer multiple of 2π radians. At these angles, the y-coordinate is -1, resulting in the minimum value of the sine function. By understanding the relationship between the angle, the unit circle, and the sine function, we can appreciate how the equation d = 0.4 sin(1760t) accurately models the oscillatory motion of the tuning fork. The trigonometric principles provide a solid foundation for understanding the behavior of sinusoidal functions and their applications in physics and engineering. In this case, it allows us to confidently determine the maximum displacement of the tuning fork based on the properties of the sine function.
Solution: Maximum Displacement is 0.4 mm
Having dissected the equation d = 0.4 sin(1760t) and explored the underlying mathematical principles, we can now confidently determine the maximum displacement of the tuning fork. As we established earlier, the maximum value of the sine function, sin(1760t), is 1. This occurs when the angle 1760t corresponds to certain points on the unit circle, specifically when the sine function reaches its peak. To find the maximum displacement, we simply substitute the maximum value of the sine function (which is 1) into the equation: d = 0.4 * sin(1760t) = 0.4 * 1 = 0.4. Therefore, the maximum displacement of the tuning fork is 0.4 millimeters. This result confirms our earlier analysis and highlights the importance of the amplitude in determining the maximum displacement of a sinusoidal motion. The amplitude, represented by the coefficient 0.4 in the equation, directly corresponds to the maximum distance the tuning fork moves from its equilibrium position. This understanding is crucial for interpreting sinusoidal equations and applying them to real-world scenarios involving oscillatory motion. Whether it's a tuning fork, a pendulum, or an alternating current circuit, the amplitude plays a central role in determining the maximum displacement or voltage. By recognizing the significance of the amplitude and the behavior of the sine function, we can easily extract valuable information from sinusoidal models and gain a deeper understanding of the underlying physics.
Conclusion: Connecting Math to the Physical World
In conclusion, our exploration of the equation d = 0.4 sin(1760t) has revealed the maximum displacement of the tuning fork to be 0.4 millimeters. This journey has not only provided a numerical answer but has also deepened our understanding of the mathematical principles governing wave motion. We've seen how a simple sinusoidal equation can effectively model the complex vibrations of a physical object like a tuning fork. By dissecting the equation, identifying key parameters like amplitude and frequency, and applying trigonometric concepts, we've connected the abstract world of mathematics to the tangible world of physics. This connection is the essence of mathematical modeling – using mathematical tools to describe and predict real-world phenomena. The example of the tuning fork serves as a powerful illustration of this concept. The equation d = 0.4 sin(1760t) is not just a string of symbols; it's a representation of a physical process, a vibration that produces sound. By understanding the equation, we gain insight into the nature of sound, the behavior of vibrating objects, and the power of mathematics to describe the world around us. This ability to connect mathematical models to physical phenomena is essential in various fields, from engineering and physics to music and acoustics. As we continue to explore the world through the lens of mathematics, we'll discover countless other examples where equations and models provide valuable insights into the workings of the universe.