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Find and graph a function of the form $\text{\hspace{0.17em}}y=a{e}^{-ct}\mathrm{cos}\left(\omega t\right)\text{\hspace{0.17em}}$ that models the information given.
Substitute the given values into the model. Recall that period is $\text{\hspace{0.17em}}\frac{2\pi}{\omega}\text{\hspace{0.17em}}$ and frequency is $\text{\hspace{0.17em}}\frac{\omega}{2\pi}.$
The following equation represents a damped harmonic motion model: $\text{\hspace{0.17em}}\text{}f\left(t\right)=5{e}^{-6t}\mathrm{cos}\left(4t\right)\text{\hspace{0.17em}}$ Find the initial displacement, the damping constant, and the frequency.
initial displacement =6, damping constant = -6, frequency = $\frac{2}{\pi}$
Find and graph a function of the form $\text{\hspace{0.17em}}y=a{e}^{-ct}\mathrm{sin}\left(\omega t\right)\text{\hspace{0.17em}}$ that models the information given.
Calculate the value of $\text{\hspace{0.17em}}\omega \text{\hspace{0.17em}}$ and substitute the known values into the model.
The damping factor is given as 10 and the amplitude is 7. Thus, the model is $\text{\hspace{0.17em}}y=7{e}^{-10t}\mathrm{sin}\left(12t\right).\text{\hspace{0.17em}}$ See [link] .
The damping factor is given as $\text{\hspace{0.17em}}0.2\text{\hspace{0.17em}}$ and the amplitude is $\text{\hspace{0.17em}}\mathrm{0.3.}\text{\hspace{0.17em}}$ The model is $\text{\hspace{0.17em}}y=0.3{e}^{-0.2t}\mathrm{sin}\left(40\pi t\right).\text{\hspace{0.17em}}$ See [link] .
Write the equation for damped harmonic motion given $\text{\hspace{0.17em}}a=10,c=0.5,$ and $\text{\hspace{0.17em}}p=2.$
$y=10{e}^{-0.5t}\mathrm{cos}\left(\pi t\right)$
A spring measuring 10 inches in natural length is compressed by 5 inches and released. It oscillates once every 3 seconds, and its amplitude decreases by 30% every second. Find an equation that models the position of the spring $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ seconds after being released.
The amplitude begins at 5 in. and deceases 30% each second. Because the spring is initially compressed, we will write A as a negative value. We can write the amplitude portion of the function as
We put $\text{\hspace{0.17em}}{\left(1-0.30\right)}^{t}\text{\hspace{0.17em}}$ in the form $\text{\hspace{0.17em}}{e}^{ct}\text{\hspace{0.17em}}$ as follows:
Now let’s address the period. The spring cycles through its positions every 3 seconds, this is the period, and we can use the formula to find omega.
The natural length of 10 inches is the midline. We will use the cosine function, since the spring starts out at its maximum displacement. This portion of the equation is represented as
Finally, we put both functions together. Our the model for the position of the spring at $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ seconds is given as
See the graph in [link] .
A mass suspended from a spring is raised a distance of 5 cm above its resting position. The mass is released at time $\text{\hspace{0.17em}}t=0\text{\hspace{0.17em}}$ and allowed to oscillate. After $\text{\hspace{0.17em}}\frac{1}{3}\text{\hspace{0.17em}}$ second, it is observed that the mass returns to its highest position. Find a function to model this motion relative to its initial resting position.
$y=5\mathrm{cos}\left(6\pi t\right)$
A guitar string is plucked and vibrates in damped harmonic motion. The string is pulled and displaced 2 cm from its resting position. After 3 seconds, the displacement of the string measures 1 cm. Find the damping constant.
The displacement factor represents the amplitude and is determined by the coefficient $\text{\hspace{0.17em}}a{e}^{-ct}\text{\hspace{0.17em}}$ in the model for damped harmonic motion. The damping constant is included in the term $\text{\hspace{0.17em}}{e}^{-ct}.\text{\hspace{0.17em}}$ It is known that after 3 seconds, the local maximum measures one-half of its original value. Therefore, we have the equation
Use algebra and the laws of exponents to solve for $\text{\hspace{0.17em}}c.$
Then use the laws of logarithms.
The damping constant is $\text{\hspace{0.17em}}\frac{\mathrm{ln}\text{\hspace{0.17em}}2}{3}.$
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