MATH 308 Lecture 16
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Quiz over 3.5 and 3.6 next Wednesday
Application: Spring Oscillation
Mass weighing 2 lbs stretches a vertical spring 6 inches (equilibrium).
Mass is pulled down 3 inches and is given an initial velocity of 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{\sqrt{3}}{8}} .
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{align} m \frac{\mathrm{d}^2 u}{\mathrm{d}t^2} &= -k \, u \\ u'' + \frac{1}{4} u &= 0 \end{align}}
At 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 = 0} , 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 u'(0) = -\frac{\sqrt{3}}{8}} . At the same 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 u(0) = -3\ \text{inches} = -\frac{1}{4}\ \text{feet}}
The solution 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 u = -\frac{1}{4} \, \sqrt{3} \, \sin{\left(\frac{1}{2} \, t \right)} - \frac{1}{4} \, \cos{\left( \frac{1}{2} \, t \right)}}
Notice the coefficients can be written as 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}{2} \left( \frac{\sqrt{3}}{2} \dots \frac{1}{2} \dots \right)} These are 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 \sin{\left(\frac{\pi}{3}\right)}} and 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 \cos{\left(\frac{\pi}{3}\right)}} , respectively. So our solution takes the form
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}{2} \left( \sin{\left( \frac{\pi}{3} \right)} \, \sin{\left(\frac{t}{2}\right)} + \cos{\left(\frac{\pi}{2}\right)} \, \cos{\left(\frac{t}{2}\right)} \right) = -\frac{1}{2} \, \cos{\left( \frac{t}{2} - \frac{\pi}{3} \right)}}
Since cosine is an even function, we can drop the negative sign: 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}{2} \, \cos{\left( \frac{t}{2} - \frac{\pi}{3} \right) }}
Some properties about this equation:
- Amplitude = 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 \tfrac{1}{2}}
- Period = 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\pi}
- Frequency = 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 \tfrac{1}{2\pi}}
In general, for any solution of the form 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 u = c_1 \, \sin{\left( \frac{t}{2} \right)} + c_2 \, \cos{\left( \frac{t}{2} \right)}}
- Amplitude = 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 \sqrt{c_1^2 + c_2^2}}
- Shift phase = 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 \tan{\phi} = \frac{c_2}{c_1}}
Suppose we have no damping and the mass is acted on by an external force of 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 2 \cos{0.4t}} pounds, the mass is at equilibrium and is released with no initial velocity.
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{\mathrm{d}^2 u}{\mathrm{d}t^2} = -\frac{1}{4}\, u + 2 \cos{0.4t}}
Maple tells us that the solution 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 u = -\frac{200}{9} \, \cos{\left(\frac{t}{2}\right)} + \frac{200}{9} \, \cos{\left(\frac{2t}{5}\right)}}
The particular solution 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 \cos{0.4t}} and the solution to the homogeneous equation 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 \cos{0.5t}}
Thanks to our trig identities, we can put this in the form 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 2 \left( \frac{200}{9} \right) \left( \sin{0.9t} \sin{0.1t} \right)} . The first sine has a period of 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 2\pi} , while the second has a period of 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 20 \pi} . These cause an interference with each other, creating a very cool-looking plot: (the blue curve is the solution, and the red curves are the 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 \pm \frac{3200}{9} \cdot 2 \sin{\left(\frac{t}{20}\right)}} amplitude and frequency envelopes:
