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.

Verified step by step guidance

1. Identify the type of motion described: since the object is attached to a coiled spring and moves up and down, this is simple harmonic motion, which can be modeled using sine or cosine functions.
2. Define the variables: let \(x(t)\) represent the distance from the rest position at time \(t\), \(A\) be the amplitude (maximum displacement), \(\omega\) be the angular frequency (related to the spring constant and mass), and \(\phi\) be the phase shift (which depends on initial conditions).
3. Write the general form of the equation for simple harmonic motion: \(x(t) = A \cos(\omega t + \phi)\) or \(x(t) = A \sin(\omega t + \phi)\).
4. For Exercise 67 (object pulled down and released), the initial displacement is maximum and velocity is zero, so use the cosine form with \(\phi = 0\), giving \(x(t) = -A \cos(\omega t)\) (negative because pulled down from rest).
5. For Exercise 68 (object propelled downward from rest position), the initial displacement is zero but initial velocity is downward, so use the sine form with appropriate phase shift, giving \(x(t) = -A \sin(\omega t)\).

Key Concepts

Simple Harmonic Motion (SHM)

Simple Harmonic Motion describes oscillatory motion where the restoring force is proportional to displacement and acts in the opposite direction. For a mass-spring system, the motion is sinusoidal, and the position varies with time as a sine or cosine function, representing periodic movement about the rest position.

Trigonometric Functions in Oscillations

Sine and cosine functions model the displacement of oscillating objects over time. The choice between sine or cosine depends on initial conditions, such as starting position or velocity. These functions capture the periodic nature of the motion with parameters for amplitude, frequency, and phase shift.

Initial Conditions and Phase Shift

Initial conditions like initial displacement and velocity determine the phase shift and amplitude in the trigonometric equation of motion. For example, pulling the object down corresponds to a nonzero initial displacement, while propelling it downward from rest position corresponds to an initial velocity, affecting the form of the solution.