Question

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.

Accepted Answer

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. 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). Write the general form of the equation for simple harmonic motion: $$x(t) = A \cos(\omega t + \phi) or$$ $$x(t) = A \sin(\omega t + \phi). $$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). 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 $$x(t) = -A \sin(\omega t).$$

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Concetti chiave

Simple Harmonic Motion (SHM)

Trigonometric Functions in Oscillations

Initial Conditions and Phase Shift