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An object moves in simple harmonic motion with amplitude $13\,\text{cm}$ and period $0.25\,\text{seconds}$ . At time $t=0\,\text{seconds}$ , its displacement $d$ from rest is $0\,\text{cm}$ , and initially it moves in a positive direction. $\newline$ Give the equation modeling the displacement $d$ as a function of time $t$ . $\newline$ $d=$ $\square$

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One-step inequalities: word problems

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Q. An object moves in simple harmonic motion with amplitude

13\,\text{cm}

and period

0.25\,\text{seconds}

. At time

t=0\,\text{seconds}

, its displacement

d

from rest is

0\,\text{cm}

, and initially it moves in a positive direction.

\newline

Give the equation modeling the displacement

d

as a function of time

t

.

\newline

d=

\square

Equation for Simple Harmonic Motion: The general form of the equation for simple harmonic motion is $d(t) = A \cdot \sin(\omega t + \varphi)$ , where $A$ is the amplitude, $\omega$ is the angular frequency, and $\varphi$ is the phase shift.
Given Amplitude: The amplitude $A$ is given as $13$ cm, so $A = 13$ .
Calculate Angular Frequency: The period $T$ is given as $0.25$ seconds. The angular frequency $\omega$ is related to the period by the formula $\omega = \frac{2\pi}{T}$ .
Calculate Phase Shift: Calculate the angular frequency: $\omega = \frac{2\pi}{0.25} = 8\pi$ .
Plug Values into Equation: Since the object starts at rest and moves in a positive direction at $t=0$ , the phase shift $\varphi$ is $0$ .
Simplify Equation: Plug the values of $A$ , $\omega$ , and $\varphi$ into the general equation: $d(t) = 13 \times \sin(8\pi t + 0)$ .
Simplify Equation: Plug the values of $A$ , $\omega$ , and $\varphi$ into the general equation: $d(t) = 13 \times \sin(8\pi t + 0)$ .Simplify the equation: $d(t) = 13 \times \sin(8\pi t)$ .

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