An object is attached to a coiled spring. The object is pulled down (negative direction from the rest position) and
then released. Write an equation for the distance of the object from its rest position after t seconds.
-amplitude $= 8 \mathrm {~cm}$; period $= 4$ seconds
A) $\mathrm { d } = - 8 \cos \frac { 1 } { 2 } \pi \mathrm { t }$
B) $d = - 4 \cos \frac { 1 } { 4 } \pi t$
C) $d = - 8 \sin \frac { 1 } { 2 } \pi t$
D) $\mathrm { d } = - 8 \cos \frac { \pi } { 4 } \mathrm { t }$

Question

Quizplus · Accepted Answer

The equation for the distance of an object attached to a spring from its rest position can be modeled by a cosine function when the initial displacement is in the negative direction. The amplitude is 8 cm, indicating the maximum displacement from the rest position. The period, which is the time it takes for one complete oscillation, is 4 seconds. The formula for the period of a cosine function is $T = \frac{2\pi}{\omega}$, where $T$ is the period and $\omega$ is the angular frequency. Solving for $\omega$ gives $\omega = \frac{2\pi}{T} = \frac{2\pi}{4} = \frac{\pi}{2}$. Therefore, the equation is $d = -8 \cos \left(\frac{1}{2}\pi t\right)$, matching choice A.

An Object Is Attached to a Coiled Spring =8 cm= 8 \mathrm {~cm}=8 cm

An Object Is Attached to a Coiled Spring $= 8 \mathrm {~cm}$