# An object moves in simple harmonic motion with period 5 seconds and amplitude 7 cm. At time displaystyle{t}={0} seconds, its displacement d from rest is -7 cm, and initially it moves in a positive direction. Give the equation modeling the displacement d as a function of time t.

An object moves in simple harmonic motion with period 5 seconds and amplitude 7 cm. At time $t=0$ seconds, its displacement d from rest is -7 cm, and initially it moves in a positive direction.
Give the equation modeling the displacement d as a function of time t.
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crocolylec
Step 1
We need to find the equation that modeling the displacement d as a function of time t for an object moving in simple harmonic motion.
Step 2
At $t=0$ seconds, the displacement d is -7 cm, and moving in a positive direction. Since initially the object is at the lowest point and moving in positive direction, so we take the negative cosine function. So the general equation for displacement as a function of time t as:
$d=-A\mathrm{cos}\left(Bt-C\right)+D$
Step 3
The amplitude is given by A, here since amplitude is 7 cm, so we have $A=7$.
The period of the object is 5 seconds, so we have
Period $=2\frac{\pi }{B}$
$5=2\frac{\pi }{B}$
$B=2\frac{\pi }{5}$
Step 4
Since the minimum displacement is at $t=0$, so there is no phase shift,
so $C=0$.
Also since there is no vertical shift, so $D=0.$
Thus the equation modeling the displacement d as a function of time t is given by:
$d=-A\mathrm{cos}\left(Bt-C\right)+D$
$=-7\mathrm{cos}\left(2\frac{\pi }{5}t-0\right)+0$
$=-7\frac{\mathrm{cos}\left(\left(2\pi t\right)\right)}{5}$
The finalli equation modeling the displacement d as a function of time t is:
$d=-7\mathrm{cos}\left(\frac{2\pi t}{5}\right)$
Jeffrey Jordon