What Is The Amplitude, Period, Frequency, And Angular Frequency Of A Body Describing SHM Given By The Expression Y = 63sin(4t)? (position In Meters, Time In Seconds)
Understanding simple harmonic motion (SHM) is crucial in various scientific fields, from physics and engineering to even aspects of biology where oscillatory systems exist. This article delves into the intricacies of SHM by analyzing a specific equation representing the motion of a body. Our focus will be on extracting key parameters such as amplitude, period, frequency, and angular frequency from the given expression. These parameters provide a comprehensive understanding of the oscillatory behavior of the system. By dissecting the equation y = 63sin(4t), we will not only determine these values but also illuminate the underlying principles that govern SHM. This exploration will equip you with the knowledge to analyze similar systems and appreciate the elegance of this fundamental type of motion. The study of simple harmonic motion begins with a clear understanding of its defining characteristics. It's a periodic motion where the restoring force is directly proportional to the displacement, leading to oscillations around an equilibrium position. This proportionality ensures that the object's acceleration is always directed towards the equilibrium point and is proportional to the displacement. This is what gives SHM its unique sinusoidal nature. The equation y = 63sin(4t) encapsulates this behavior, where 'y' represents the displacement at any given time 't'. Our task is to decipher the information encoded within this equation, specifically the amplitude, period, frequency, and angular frequency. Each of these parameters provides a different perspective on the motion, allowing us to fully characterize the oscillation. For instance, the amplitude tells us the maximum displacement from the equilibrium, while the period dictates the time taken for one complete oscillation. The frequency, conversely, indicates how many oscillations occur per unit time, and the angular frequency relates to the rate of change of the phase of the oscillation. Understanding these parameters is not just an academic exercise; it has practical implications in various applications, from designing pendulum clocks to analyzing molecular vibrations. By mastering the interpretation of equations like y = 63sin(4t), you gain a powerful tool for understanding and predicting the behavior of oscillatory systems.
Deciphering the Equation: y = 63sin(4t)
The equation y = 63sin(4t) is the key to unlocking the characteristics of this specific SHM. In this equation, 'y' represents the displacement of the body from its equilibrium position at time 't'. The equation is structured in the standard form of a sinusoidal function, which is typical for SHM. By comparing this equation to the general form of the SHM equation, we can directly identify the values of amplitude and angular frequency. The general form of the SHM equation is often written as y = A sin(ωt + φ), where 'A' represents the amplitude, 'ω' represents the angular frequency, 't' is time, and 'φ' is the phase constant. In our case, y = 63sin(4t), we can see a clear resemblance to the general form. By direct comparison, we can identify that the amplitude (A) is 63 meters. This value signifies the maximum displacement of the body from its equilibrium position during its oscillation. It's the peak value the displacement reaches in either direction. The angular frequency (ω) is 4 radians per second. This parameter is crucial as it dictates the rate at which the oscillation occurs. It's not the same as the frequency (f), which we will calculate later, but it's directly related to it. The angular frequency essentially tells us how quickly the phase of the oscillation changes over time. The phase constant (φ) in our equation is 0, which means the oscillation starts at the equilibrium position. If there were a non-zero phase constant, it would indicate a shift in the starting point of the oscillation. Understanding the components of this equation is essential for calculating the other parameters we're interested in, namely the period and frequency. The amplitude and angular frequency are the building blocks upon which we can determine the time it takes for one complete oscillation and the number of oscillations per unit time. The careful analysis of the equation y = 63sin(4t) provides us with a solid foundation for understanding the motion it describes. It's a testament to the power of mathematical representation in capturing the essence of physical phenomena.
Determining the Amplitude: The Maximum Displacement
To understand the motion fully, we must first determine the amplitude. In the equation y = 63sin(4t), the amplitude is the coefficient of the sine function. This value directly represents the maximum displacement of the body from its equilibrium position. In this case, the amplitude is 63 meters. This means the body oscillates back and forth, reaching a maximum displacement of 63 meters in either direction from its resting position. The amplitude is a crucial parameter as it defines the extent of the oscillation. A larger amplitude indicates a more energetic oscillation, where the body travels further from its equilibrium point. Conversely, a smaller amplitude signifies a gentler oscillation with less displacement. The amplitude is not only a descriptive parameter but also a physical one. It's directly related to the energy of the oscillating system. In the context of SHM, the total energy of the system is proportional to the square of the amplitude. This means that doubling the amplitude quadruples the energy of the oscillation. This relationship highlights the significance of amplitude in understanding the energetics of SHM. For example, in a pendulum, a larger amplitude swing requires more energy input. Similarly, in a vibrating string, a larger amplitude corresponds to a louder sound, which carries more energy. The amplitude is also important in practical applications of SHM. In designing oscillators, controlling the amplitude is crucial for achieving the desired performance. For instance, in electronic circuits, oscillators are used to generate signals of specific frequencies and amplitudes. The amplitude of these signals must be carefully controlled to ensure proper functioning of the circuit. Understanding the concept of amplitude is fundamental to grasping the behavior of SHM. It's a parameter that not only describes the motion but also provides insights into the energy and practical applications of oscillatory systems.
Calculating the Period: The Time for One Oscillation
Moving on from amplitude, calculating the period of oscillation is the next crucial step. The period (T) is defined as the time it takes for one complete cycle of the motion. In SHM, this is the time it takes for the body to return to its initial position and velocity. The period is inversely related to the angular frequency (ω), which we identified as 4 radians per second in the equation y = 63sin(4t). The relationship between period and angular frequency is given by the formula: T = 2π/ω. Substituting the value of ω, we get T = 2π/4 = π/2 seconds. This is the time it takes for the body to complete one full oscillation. The period is a fundamental parameter that characterizes the time scale of the oscillation. A shorter period means the oscillation is faster, while a longer period indicates a slower oscillation. The period is not just a mathematical quantity; it has a direct physical interpretation. It's the time interval between successive peaks or troughs in the displacement of the oscillating body. For instance, if we were to observe the motion graphically, the period would be the distance along the time axis between two consecutive peaks in the sine wave. Understanding the period is essential in various applications of SHM. For example, in the design of clocks, the period of the pendulum or the balance wheel determines the accuracy of the timekeeping. Similarly, in musical instruments, the period of vibration of the strings or air columns determines the pitch of the sound produced. The period is also important in understanding resonance phenomena. Resonance occurs when an external force is applied to an oscillating system at a frequency close to its natural frequency (which is inversely proportional to the period). This can lead to a large amplitude response, which can be either beneficial or detrimental depending on the application. The calculation of the period from the angular frequency is a key step in characterizing SHM. It provides a measure of the time scale of the oscillation and is crucial in understanding the behavior and applications of oscillatory systems.
Finding the Frequency: Oscillations per Unit Time
Next, let's delve into the frequency of the motion. The frequency (f) represents the number of complete oscillations that occur per unit of time, typically measured in Hertz (Hz), where 1 Hz is equal to one oscillation per second. The frequency is inversely related to the period (T), which we calculated earlier. The relationship is expressed by the formula: f = 1/T. Since we found the period to be π/2 seconds, the frequency is f = 1/(π/2) = 2/π Hz. This value represents how many full cycles of oscillation the body completes in one second. A higher frequency indicates a more rapid oscillation, while a lower frequency signifies a slower oscillation. The frequency is a crucial parameter in many physical phenomena. In the context of sound, frequency corresponds to pitch; a higher frequency sound is perceived as higher pitched, and a lower frequency sound is perceived as lower pitched. In electromagnetic waves, frequency determines the color of light; higher frequency light is blue or violet, while lower frequency light is red. In electrical circuits, frequency is a key parameter in alternating current (AC) systems, where it determines the rate at which the current changes direction. Understanding the frequency of SHM is essential in various applications. In the design of musical instruments, the frequency of vibration of the strings, air columns, or membranes determines the notes produced. In communication systems, radio waves and microwaves are used to transmit information, and their frequencies are carefully chosen to avoid interference. In medical imaging, ultrasound waves are used to create images of internal organs, and the frequency of these waves affects the resolution and penetration depth of the images. The frequency, along with the period, provides a complete picture of the time-dependent behavior of SHM. While the period tells us the time for one oscillation, the frequency tells us how many oscillations occur in a given time interval.
Angular Frequency: The Rate of Phase Change
Finally, we turn our attention to the angular frequency, a key parameter in understanding SHM. As previously identified from the equation y = 63sin(4t), the angular frequency (ω) is 4 radians per second. The angular frequency is not simply another measure of how fast the oscillation is; it provides a deeper insight into the rate of change of the phase of the oscillation. To understand this, consider the sinusoidal nature of SHM. The sine function oscillates between -1 and 1, and the argument of the sine function (4t in this case) determines the phase of the oscillation. The angular frequency tells us how quickly this phase changes with time. In other words, it's the rate at which the angle in the sine function is changing. The unit of angular frequency is radians per second, which reflects this rate of change of angle. The angular frequency is related to the ordinary frequency (f) by the formula: ω = 2πf. This relationship highlights the connection between the rate of oscillation and the rate of change of phase. A higher angular frequency means a faster oscillation and a more rapid change in phase. The angular frequency is particularly useful in analyzing the energy of SHM. The total energy of the system is proportional to the square of the angular frequency. This means that doubling the angular frequency quadruples the energy of the oscillation, assuming the amplitude remains constant. The angular frequency is also important in understanding the resonance phenomena we discussed earlier. The natural frequency of an oscillating system, which determines its resonant behavior, is directly related to its angular frequency. Systems tend to resonate when driven at a frequency close to their natural frequency, and this natural frequency is determined by the angular frequency. Understanding the angular frequency provides a more complete picture of SHM. It's not just a measure of how fast the oscillation is; it's also a measure of the rate of change of phase and is closely related to the energy and resonant behavior of the system.
Conclusion: The Symphony of SHM Parameters
In conclusion, by analyzing the equation y = 63sin(4t), we have successfully determined the key parameters that characterize this simple harmonic motion. The amplitude, period, frequency, and angular frequency paint a comprehensive picture of the oscillatory behavior of the body. The amplitude of 63 meters tells us the maximum displacement from equilibrium. The period of π/2 seconds reveals the time for one complete oscillation. The frequency of 2/π Hz indicates the number of oscillations per second. And the angular frequency of 4 radians per second describes the rate of change of the phase of the oscillation. These parameters are not just isolated values; they are interconnected and provide a holistic understanding of SHM. The amplitude defines the extent of the motion, the period and frequency dictate the time scale of the oscillation, and the angular frequency provides insight into the rate of change of phase and the energy of the system. The ability to extract these parameters from an equation like y = 63sin(4t) is a powerful tool in analyzing oscillatory systems. It allows us to predict the behavior of these systems, design oscillators for various applications, and understand phenomena like resonance. Simple harmonic motion is a fundamental concept in physics and engineering, and its understanding is crucial for various applications. From the oscillations of a pendulum to the vibrations of atoms in a crystal, SHM is a ubiquitous phenomenon in nature. By mastering the analysis of SHM, you gain a deeper appreciation for the world around you and the principles that govern its behavior. The journey of understanding SHM through the analysis of equations like y = 63sin(4t) is a rewarding one. It's a testament to the power of mathematics in describing and explaining the physical world. The symphony of parameters – amplitude, period, frequency, and angular frequency – harmoniously combine to reveal the elegant dance of simple harmonic motion.