Equation For Simple Harmonic Motion Amplitude 6 Inches, Frequency 6 Cycles Per Second

Understanding Simple Harmonic Motion

When delving into the world of physics, understanding simple harmonic motion (SHM) is crucial. This type of motion is characterized by a restoring force that is directly proportional to the displacement and acts in the opposite direction. Imagine a pendulum swinging back and forth or a mass attached to a spring oscillating up and down; these are classic examples of SHM. To fully grasp SHM, we need to understand key concepts such as amplitude, frequency, and period, which play pivotal roles in defining the motion's characteristics. Amplitude, in this context, represents the maximum displacement from the equilibrium position. It's the measure of how far the object moves from its resting point. A larger amplitude indicates a more extensive oscillation. Frequency, on the other hand, quantifies how many complete cycles of the motion occur per unit of time, typically measured in Hertz (Hz), which corresponds to cycles per second. A higher frequency means the object oscillates more rapidly. Lastly, the period is the time it takes for one complete cycle of the motion to occur, and it is inversely related to the frequency. Understanding these components allows us to accurately describe and predict the behavior of objects undergoing simple harmonic motion. Now, let's dive deeper into formulating the equation that mathematically represents this fascinating phenomenon.

In our given scenario, we have an object undergoing simple harmonic motion with a specified amplitude and frequency. The amplitude is given as 6 inches, which tells us the maximum displacement of the object from its equilibrium position. This value is essential as it directly influences the magnitude of the motion. The frequency is given as 6 cycles per second, indicating that the object completes six full oscillations every second. This high frequency suggests a rapid back-and-forth movement. Furthermore, the problem states that the displacement is zero at time t=0, which provides a crucial initial condition for our equation. This condition implies that the object starts its motion at the equilibrium position. With these parameters in hand—amplitude, frequency, and the initial condition—we can construct the equation that precisely describes the simple harmonic motion in question. The equation will capture the oscillatory nature of the motion, reflecting how the displacement changes over time. The next step involves using these values to determine the specific parameters within the general equation for SHM, ultimately leading us to a clear and concise mathematical representation of this particular motion. This representation will not only allow us to visualize the motion but also to predict the object's position at any given time.

The general equation for simple harmonic motion is typically expressed using trigonometric functions, either sine or cosine, as these functions inherently describe oscillatory behavior. The choice between sine and cosine often depends on the initial conditions of the motion. Given that the displacement is zero at t=0, the sine function is the more appropriate choice because sin(0) = 0, satisfying the initial condition. The general form of the equation using the sine function is: y(t) = A * sin(ωt + φ), where:

• y(t) represents the displacement at time t.
• A is the amplitude of the motion.
• ω (omega) is the angular frequency.
• t is the time.
• φ (phi) is the phase constant.

In our specific case, we are given the amplitude A = 6 inches. The angular frequency ω is related to the frequency f (in Hz) by the equation ω = 2πf. Since the frequency is 6 cycles per second, the angular frequency is ω = 2π * 6 = 12π radians per second. Now, considering the initial condition that the displacement is zero at t=0, we can determine the phase constant φ. Plugging in t=0 and y(0)=0 into the general equation, we get 0 = 6 * sin(12π * 0 + φ), which simplifies to sin(φ) = 0. The solutions for φ where sin(φ) = 0 are φ = 0, π, 2π, and so on. However, since we are starting at the equilibrium position and the motion begins in the positive direction, we can take φ = 0. This simplifies our equation and provides a clear starting point for describing the motion. By substituting the known values of amplitude and angular frequency into the general equation, we arrive at the specific equation that accurately models the simple harmonic motion described in the problem statement. This equation will allow us to calculate the displacement of the object at any given time, offering a comprehensive understanding of its oscillatory behavior.

Deriving the Equation for the Given Conditions

To derive the equation for simple harmonic motion under the given conditions, we start with the general form of the equation, which, as established, is most appropriately represented by a sine function due to the initial condition of zero displacement at t=0. The general equation is: y(t) = A * sin(ωt + φ). We know the amplitude (A) is 6 inches and the frequency (f) is 6 cycles per second. We have also determined that the phase constant (φ) is 0. The key missing piece is the angular frequency (ω), which we can calculate using the relationship ω = 2πf.

Substituting the given frequency f = 6 cycles per second into the formula ω = 2πf, we find ω = 2π * 6 = 12π radians per second. This value represents the rate at which the object oscillates in radians per unit time. Now that we have calculated the angular frequency, we can substitute all the known values—amplitude, angular frequency, and phase constant—into the general equation for simple harmonic motion. This substitution will yield the specific equation that describes the motion under the given conditions. The equation will capture the oscillatory behavior, reflecting the periodic nature of the motion and how the displacement changes over time. By carefully substituting each value, we ensure that the equation accurately represents the motion's characteristics, allowing us to predict the object's position at any given time. This process of deriving the specific equation from the general form highlights the importance of understanding the relationship between the physical parameters of the motion and their mathematical representation. The resulting equation will be a concise and powerful tool for analyzing and understanding the simple harmonic motion in question.

Substituting A = 6 inches, ω = 12π radians per second, and φ = 0 into the general equation, we get:

y(t) = 6 * sin(12πt). This equation represents the displacement y(t) in inches at any time t in seconds. It clearly shows the sinusoidal nature of the motion, with the displacement oscillating between -6 inches and +6 inches. The 12π term inside the sine function dictates the frequency of the oscillation, ensuring that the motion completes 6 cycles every second. This equation is the culmination of our analysis, providing a precise mathematical description of the simple harmonic motion under the given conditions. It encapsulates all the essential information—amplitude, frequency, and initial phase—into a single, elegant expression. This equation is not just a theoretical construct; it is a practical tool that allows us to calculate the position of the object at any point in time, making it invaluable for understanding and predicting the behavior of the system. The process of deriving this equation demonstrates the power of mathematical modeling in physics, allowing us to translate real-world phenomena into precise mathematical representations.

The Final Equation and Its Implications

The final equation that describes the simple harmonic motion with an amplitude of 6 inches and a frequency of 6 cycles per second, given zero displacement at t=0, is:

y(t) = 6sin(12πt)

This equation encapsulates all the information we have gathered and provides a complete mathematical representation of the motion. The sine function signifies the oscillatory nature, and the parameters within the function define the specific characteristics of the oscillation. The coefficient 6 represents the amplitude, which is the maximum displacement from the equilibrium position. This value indicates that the object moves as far as 6 inches away from its resting position in either direction. The term 12π is the angular frequency, which is directly related to the frequency of oscillation. It determines how rapidly the motion oscillates back and forth. The argument of the sine function, 12πt, shows how the displacement changes with time. The equation allows us to calculate the displacement y(t) at any given time t, providing a clear picture of the object's position as it oscillates. This is a powerful tool for predicting and analyzing the motion. Furthermore, the equation confirms that at t=0, the displacement y(0) is indeed zero, as sin(0) = 0. This validates our initial condition and ensures that the equation accurately represents the given scenario.

This equation not only describes the motion mathematically but also provides insights into the physical behavior of the oscillating object. It highlights the periodic nature of simple harmonic motion, where the motion repeats itself after a fixed interval. The equation also demonstrates the relationship between the amplitude and the energy of the system. A larger amplitude would imply a greater energy in the system, as the object moves farther from its equilibrium position. The angular frequency, on the other hand, is related to the stiffness of the restoring force. A higher angular frequency indicates a stronger restoring force, causing the object to oscillate more rapidly. The simplicity of the equation belies its power in describing a wide range of physical phenomena, from the motion of a pendulum to the vibrations of atoms in a solid. The equation serves as a fundamental building block in understanding more complex oscillatory systems. It demonstrates the elegance and efficiency of mathematical descriptions in capturing the essence of physical processes. By understanding and manipulating this equation, we can gain a deeper appreciation for the underlying principles of simple harmonic motion and its applications in various fields of science and engineering. The derived equation is a testament to the power of mathematical modeling in physics, allowing us to predict and analyze the behavior of oscillating systems with precision and clarity. The final equation is a concise yet comprehensive representation of the motion, allowing us to understand and predict the object's position at any given time.