A pendulum is swinging next to a wall. $\newline$ The distance $D(t)$ (in $\mathrm{cm}$ ) between the bob of the pendulum and the wall as a function of time $t$ (in seconds) can be modeled by a sinusoidal expression of the form $a \cdot \sin (b \cdot t)+d$ . $\newline$ At $t=0$ , when the pendulum is exactly in the middle of its swing, the bob is $5 \mathrm{~cm}$ away from the wall. The bob reaches the closest point to the wall, which is $3 \mathrm{~cm}$ from the wall, $1$ second later. $\newline$ Find $D(t)$ . $\newline$ $t$ should be in radians. $\newline$ $D(t)=\square$

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Q. A pendulum is swinging next to a wall.

\newline

The distance

D(t)

(in

\mathrm{cm}

) between the bob of the pendulum and the wall as a function of time

t

(in seconds) can be modeled by a sinusoidal expression of the form

a \cdot \sin (b \cdot t)+d

\newline

t=0

, when the pendulum is exactly in the middle of its swing, the bob is

5 \mathrm{~cm}

away from the wall. The bob reaches the closest point to the wall, which is

3 \mathrm{~cm}

from the wall,

1

second later.

\newline

Find

D(t)

\newline

t

should be in radians.

\newline

D(t)=\square

Initial Position Calculation: The pendulum is in the middle of its swing at $t=0$ , so the distance from the wall is the average of the maximum and minimum distances. This is the value of $d$ in the equation.
Amplitude Calculation: Since the pendulum is $5\,\text{cm}$ from the wall at $t=0$ , $d=5$ .
Period Calculation: One second later, the pendulum is at its closest point to the wall, which is $3\,\text{cm}$ . This means the amplitude $a$ is the difference between the average distance and the closest distance, so $a=5\,\text{cm} - 3\,\text{cm} = 2\,\text{cm}$ .
Angular Frequency Calculation: The pendulum reaches its closest point to the wall $1$ second later, which is a quarter of the period of the sinusoidal function. Therefore, the period $T$ is $4$ seconds.
Sinusoidal Function Determination: To convert the period $T$ into radians, we use the formula $T = 2\pi / b$ , where $b$ is the angular frequency in radians per second. Solving for $b$ gives us $b = 2\pi / T$ .
Sinusoidal Function Determination: To convert the period $T$ into radians, we use the formula $T=2\pi/b$ , where $b$ is the angular frequency in radians per second. Solving for $b$ gives us $b=2\pi/T$ .Substitute $T$ with $4$ seconds to find $b$ : $b=2\pi/4=\pi/2$ radians per second.
Sinusoidal Function Determination: To convert the period $T$ into radians, we use the formula $T=2\pi/b$ , where $b$ is the angular frequency in radians per second. Solving for $b$ gives us $b=2\pi/T$ .Substitute $T$ with $4$ seconds to find $b$ : $b=2\pi/4=\pi/2$ radians per second.Now we have all the parameters for the sinusoidal function: $a=2$ cm, $T=2\pi/b$ $0$ radians per second, and $T=2\pi/b$ $1$ cm. The function $T=2\pi/b$ $2$ is therefore $T=2\pi/b$ $3$ .