How To Determine Amplitude From A Displacement Equation In Oscillatory Motion

Introduction

In the realm of physics, oscillatory motion is a fundamental concept that describes the repetitive back-and-forth movement of an object around a central equilibrium position. This type of motion is ubiquitous in nature, from the swinging of a pendulum to the vibrations of atoms in a solid. Understanding the characteristics of oscillatory motion is crucial for comprehending various physical phenomena. Among these characteristics, amplitude stands out as a key parameter that defines the extent of the displacement from the equilibrium position. In this comprehensive guide, we will delve into the concept of amplitude, its significance in oscillatory motion, and how to determine it from a given equation. We will explore the mathematical representation of oscillatory motion and demonstrate how to extract the amplitude from such representations. Furthermore, we will discuss the relationship between amplitude and the energy of an oscillating system. By the end of this guide, you will have a thorough understanding of amplitude and its role in characterizing oscillatory motion.

Defining Oscillatory Motion and Its Significance

Oscillatory motion, also known as periodic motion, is the repetitive movement of an object or system between two or more extreme positions. This type of motion is characterized by a restoring force that pulls the object back towards its equilibrium position. The interplay between this restoring force and the object's inertia results in the oscillation. Oscillatory motion is not merely a theoretical concept; it manifests in a wide array of physical systems. Consider the simple pendulum, swinging rhythmically back and forth under the influence of gravity. Or the mass-spring system, where a mass attached to a spring oscillates due to the restoring force of the spring. Even at the atomic level, atoms in a solid vibrate around their equilibrium positions, exhibiting oscillatory motion. The study of oscillatory motion is essential in various fields of physics and engineering. In mechanics, it helps us understand the behavior of vibrating systems, such as machines and structures. In acoustics, it is the basis for sound waves and their propagation. In electromagnetism, oscillatory motion plays a crucial role in the generation and transmission of electromagnetic waves. Furthermore, the principles of oscillatory motion are applied in the design of various devices, including clocks, musical instruments, and electronic circuits.

Introducing Amplitude: A Key Parameter

Amplitude, in the context of oscillatory motion, is the maximum displacement of an object from its equilibrium position during its oscillation. It is a measure of the intensity or magnitude of the oscillation. The larger the amplitude, the greater the displacement from equilibrium, and the more energy the system possesses. Amplitude is typically measured in units of length, such as meters (m) or centimeters (cm). It is a positive quantity, as it represents a distance. Consider a simple pendulum. Its amplitude is the maximum angular displacement from its vertical equilibrium position. For a mass-spring system, the amplitude is the maximum displacement of the mass from its resting position. In the case of a sound wave, the amplitude corresponds to the maximum change in pressure from the ambient pressure. Amplitude is a critical parameter in characterizing oscillatory motion because it directly relates to the energy of the system. For a simple harmonic oscillator, such as a mass-spring system, the energy is proportional to the square of the amplitude. This means that doubling the amplitude quadruples the energy of the oscillation. Amplitude also influences other properties of oscillatory motion, such as the period and frequency. While the period (the time for one complete oscillation) and frequency (the number of oscillations per unit time) are independent of amplitude for simple harmonic motion, this is not always the case for more complex oscillations. Understanding amplitude is essential for analyzing and predicting the behavior of oscillatory systems. It allows us to quantify the intensity of the oscillation and relate it to other physical quantities, such as energy and frequency.

Problem Statement: Determining Amplitude from an Equation

Let's consider the specific problem presented: The displacement y in centimeters (cm) is given in terms of time t in seconds (s) by the equation: y = 3sin(3.14t) + 4cos(3.14t). The task is to determine the amplitude of this oscillatory motion. This problem exemplifies a common scenario in physics where we need to extract information about the motion from its mathematical representation. The equation y = 3sin(3.14t) + 4cos(3.14t) describes a sinusoidal oscillation, which is a fundamental type of oscillatory motion. Sinusoidal oscillations are characterized by their smooth, repetitive pattern, and they can be described using sine and cosine functions. The equation given is a superposition of two sinusoidal functions: 3sin(3.14t) and 4cos(3.14t). This means that the motion is a combination of two oscillations with different amplitudes and phases. To find the amplitude of the combined motion, we cannot simply add the amplitudes of the individual oscillations. Instead, we need to use a mathematical technique to combine the two sinusoidal functions into a single sinusoidal function. This technique involves using trigonometric identities to rewrite the equation in the form y = Asin(ωt + φ), where A is the amplitude, ω is the angular frequency, and φ is the phase constant. Once we have the equation in this form, the amplitude A can be directly read off. Solving this problem requires a solid understanding of trigonometry and the properties of sinusoidal functions. It also demonstrates the importance of mathematical tools in analyzing physical phenomena.

Rewriting the Equation: Combining Sinusoidal Functions

To determine the amplitude, we need to rewrite the given equation y = 3sin(3.14t) + 4cos(3.14t) in the standard form of a sinusoidal function, which is y = Asin(ωt + φ). This involves combining the two sinusoidal terms into a single term using trigonometric identities. The key identity we will use is the sine addition formula: sin(a + b) = sin(a)cos(b) + cos(a)sin(b). We want to find an amplitude A and a phase constant φ such that: Asin(3.14t + φ) = 3sin(3.14t) + 4cos(3.14t). Expanding the left side using the sine addition formula, we get: A[sin(3.14t)cos(φ) + cos(3.14t)sin(φ)] = 3sin(3.14t) + 4cos(3.14t). Now, we can equate the coefficients of the sin(3.14t) and cos(3.14t) terms on both sides: Acos(φ) = 3 and Asin(φ) = 4. To find A, we can square both equations and add them together: (Acos(φ))^2 + (Asin(φ))^2 = 3^2 + 4^2. This simplifies to: A^2(cos2(φ) + sin^2(φ)) = 9 + 16. Since cos^2(φ) + sin^2(φ) = 1, we have: A^2 = 25. Taking the square root of both sides, we get: A = 5. (We take the positive root since amplitude is a positive quantity.) To find the phase constant φ, we can divide the equation Asin(φ) = 4 by the equation Acos(φ) = 3: tan(φ) = 4/3. Therefore, φ = arctan(4/3). However, we do not need to find the value of φ to determine the amplitude. The amplitude A is the coefficient of the sinusoidal function, which we have found to be 5.

Determining the Amplitude: The Solution

From the previous steps, we have determined that the amplitude A of the oscillatory motion described by the equation y = 3sin(3.14t) + 4cos(3.14t) is 5 cm. This means that the maximum displacement of the object from its equilibrium position is 5 cm. The amplitude is a crucial parameter that characterizes the intensity of the oscillation. A larger amplitude indicates a greater displacement and, consequently, a higher energy associated with the oscillation. In this case, the amplitude of 5 cm tells us that the object oscillates with a significant displacement from its equilibrium position. The other options provided, 3 cm, 4 cm, and 7 cm, are incorrect. The amplitude is not simply the sum of the coefficients of the sine and cosine terms (3 + 4 = 7) because the two oscillations are out of phase. Instead, we need to combine them using trigonometric identities, as we have demonstrated. The amplitude of 5 cm represents the magnitude of the combined oscillation, taking into account the phase difference between the two sinusoidal components. This problem highlights the importance of understanding how to manipulate trigonometric functions and apply trigonometric identities to solve physics problems. It also reinforces the concept of amplitude as a fundamental characteristic of oscillatory motion.

Conclusion

In conclusion, the amplitude of the oscillatory motion described by the equation y = 3sin(3.14t) + 4cos(3.14t) is 5 cm. This was determined by rewriting the equation in the standard sinusoidal form using trigonometric identities. The amplitude represents the maximum displacement from the equilibrium position and is a key parameter in characterizing oscillatory motion. This problem demonstrates the importance of understanding trigonometric functions and their applications in physics. Amplitude, as we have explored, is a fundamental concept in the study of oscillatory motion. It provides a measure of the intensity of the oscillation and is directly related to the energy of the system. Understanding amplitude is crucial for analyzing and predicting the behavior of oscillating systems in various fields of physics and engineering. From simple pendulums to complex vibrations in molecules, amplitude plays a vital role in determining the characteristics of these motions. This guide has provided a comprehensive explanation of amplitude, its significance, and how to determine it from a given equation. By mastering these concepts, you will be well-equipped to tackle a wide range of problems involving oscillatory motion.