In Exercises 67–68, an object is attached to a coiled spring. In Exercise 67, the object is pulled down (negative direction from the rest position) and then released. In Exercise 68, the object is propelled downward from its rest position. Write an equation for the distance of the object from its rest position after t seconds.

Key Concepts

Simple Harmonic Motion

Simple Harmonic Motion (SHM) describes the oscillatory motion of an object attached to a spring, where the restoring force is proportional to the displacement from its equilibrium position. This motion can be modeled using sine or cosine functions, which represent the periodic nature of the oscillation. Understanding SHM is crucial for deriving the equations that describe the distance of the object from its rest position over time.

Differential Equations

Differential equations are mathematical equations that relate a function to its derivatives. In the context of SHM, they can be used to model the motion of the spring system, leading to equations that describe the position of the object as a function of time. Solving these equations helps in determining the specific form of the distance equation based on initial conditions, such as the initial displacement and velocity.

Phase Shift

Phase shift refers to the horizontal shift in the graph of a periodic function, such as sine or cosine. In the context of the problem, the initial conditions of the object's motion (whether it is pulled down or propelled downward) will affect the phase of the resulting trigonometric function. Understanding phase shifts is essential for accurately writing the equation that represents the distance of the object from its rest position after a given time.