Understanding the Maximum Displacement in Simple Harmonic Motion

Simple harmonic motion (SHM) is a fundamental concept in physics, commonly described by trigonometric functions. The equation y Asin(ωt - φ) represents a displaced oscillating system, where A is the amplitude, ω is the angular frequency, t is the time, and φ is the phase angle. This article aims to explain the calculation of the maximum displacement of SHM, specifically with the application of y Asin(10t - π/6).

Maximum Displacement in SHM

In simple harmonic motion, the displacement y from the equilibrium position at any time t is described by the equation ( y A sin(omega t - phi) ). The key parameters are the amplitude A and the phase angle φ. The maximum displacement from the starting reference point at time t 0 can be calculated by analyzing the function at this point.

At t 0, the value of y is:

y A sin ( 0 - π 6 ) - 1 / 2 A A - A 2

This result shows that at t 0, the displacement is (-frac{A}{2} ). The maximum and minimum values of y over time are A and -A, respectively. Therefore, the maximum displacement from its position at time t 0 is the difference between the amplitude A and the negative half of the amplitude, which is:

Δy A - A 2 A 2

Thus, the maximum displacement from its starting reference point is (frac{3A}{2} ), considering the absolute difference from (-frac{A}{2}) to A.

Application in Real-Life Scenario

Simple harmonic motion can be observed in various real-life applications, such as the motion of a pendulum, sound waves, and electromagnetic waves. For instance, in the context of a pendulum, the amplitude A can represent the maximum angular displacement from the vertical. Understanding the maximum displacement is crucial for analyzing the system’s behavior and predicting its future motion.

Conclusion

In summary, the maximum displacement in simple harmonic motion represented by y Asin(10t - π/6) at t 0 is (-frac{A}{2} ), and the maximum displacement from the starting reference point is (frac{3A}{2} ). This understanding is vital for accurately modeling and predicting the behavior of oscillating systems in various fields, including physics, engineering, and applied sciences.

In conclusion, understanding simple harmonic motion and its parameters is essential to describing and predicting the behavior of oscillating systems. The calculation of maximum displacement provides a clear picture of the system's motion and helps in various applications.