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 $= 14$ in.; period $= 6$ seconds
A) $d = - 14 \cos \frac { 1 } { 3 } \pi t$
B) $\mathrm { d } = - 6 \cos \frac { 1 } { 7 } \pi \mathrm { t }$
C) $d = - 14 \sin \frac { 1 } { 3 } \pi t$
D) $d = - 14 \cos \frac { \pi } { 6 } t$

Question

Quizplus · Accepted Answer

The correct equation for the motion of the object is $d = - 14 \cos \frac { 1 } { 3 } \pi t$. This is because the amplitude of the motion is 14 inches, indicating the maximum displacement from the rest position, and the period is 6 seconds, which means the motion repeats every 6 seconds. The negative sign indicates that the object starts in the negative direction from the rest position. The formula for the period $T$ of a cosine function is $T = \frac{2\pi}{\omega}$, where $\omega$ is the angular frequency. Given $T = 6$, we solve for $\omega$ to get $\omega = \frac{2\pi}{6} = \frac{\pi}{3}$, which matches the equation in choice A.

An Object Is Attached to a Coiled Spring =14= 14=14 In; Period

An Object Is Attached to a Coiled Spring $= 14$ In; Period