009A Sample Midterm 1, Problem 5

The displacement from equilibrium of an object in harmonic motion on the end of a spring is:

$y={\frac {1}{3}}\cos(12t)-{\frac {1}{4}}\sin(12t)$

where $y$ is measured in feet and $t$ is the time in seconds.

Determine the position and velocity of the object when $t={\frac {\pi }{8}}.$

Foundations:
What is the relationship between position Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle s(t)} and velocity Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle v(t)} of an object?
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle v(t)=s'(t)}

Solution:

Step 1:
To find the position of the object at $t={\frac {\pi }{8}},$
we need to plug $t={\frac {\pi }{8}}$ into the equation Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y.}
Thus, we have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{array}{rcl}\displaystyle {y{\bigg (}{\frac {\pi }{8}}{\bigg )}}&=&\displaystyle {{\frac {1}{3}}\cos {\bigg (}{\frac {12\pi }{8}}{\bigg )}-{\frac {1}{4}}\sin {\bigg (}{\frac {12\pi }{8}}{\bigg )}}\\&&\\&=&\displaystyle {{\frac {1}{3}}\cos {\bigg (}{\frac {3\pi }{2}}{\bigg )}-{\frac {1}{4}}\sin {\bigg (}{\frac {3\pi }{2}}{\bigg )}}\\&&\\&=&\displaystyle {0-{\frac {1}{4}}(-1)}\\&&\\&=&\displaystyle {{\frac {1}{4}}{\text{ foot}}.}\end{array}}}

Step 2:
Now, to find the velocity function, we need to take the derivative of the position function.
Thus, we have
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{v(t)} & = & \displaystyle{y'}\\ &&\\ & = & \displaystyle{\frac{-1}{3}\sin(12t)(12)-\frac{1}{4}\cos(12t)(12)}\\ &&\\ & = & \displaystyle{-4\sin(12t)-3\cos(12t).} \end{array}}
Therefore, the velocity of the object at time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t=\frac{\pi}{8}} is
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{v\bigg(\frac{\pi}{8}\bigg)} & = & \displaystyle{-4\sin\bigg(\frac{3\pi}{2}\bigg)-3\cos\bigg(\frac{3\pi}{2}\bigg)}\\ &&\\ & = & \displaystyle{-4(-1)+0}\\ &&\\ & = & \displaystyle{4 \text{ feet/second}.} \end{array}}

Final Answer:
position is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{4} \text{ foot}.}
velocity is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4 \text{ feet/second}.}

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