Example: The displacement of a mass suspended by a spring is modeled by the equation:
\[ y = -12\sin(6\pi t) \]where \(y\) is measured in cm and \(t\) in seconds.
- Find the amplitude, period, and frequency of the motion of the mass.
- Sketch a graph of the displacement of the mass from its resting position.
Solution Based on the equation \(y = -12\sin(6\pi t)\), the amplitude is 12 cm, the period is \(P=\frac{2\pi}{6\pi}=\frac{{1}}{{3}}\) and the frequency is \(f = \frac{{1}}{{P}} = 3\) cycles per second (hz).
To determine each of the critical input values, we look at the quarter period: \(\frac{{P}}{{4}} = \frac{{1}}{{3}}\times \frac{{1}}{{4}}=\frac{{1}}{{12}}\). Then we can write down the input values as the first few multiples of the quarter period:
\(t = \frac{{1}}{{12}},\;\;\frac{{2}}{{12}},\;\;\frac{{3}}{{12}},\;\;\frac{{4}}{{12}},\;\;\frac{{5}}{{12}},\dots\)
Of course, we should simplify these fractions, but by counting up with the unsimplified version, it is much easier. Simplified: \(t = \frac{{1}}{{12}},\;\;\frac{{1}}{{6}},\;\;\frac{{1}}{{4}},\;\;\frac{{1}}{{3}},\;\;\frac{{5}}{{12}},\dots\) and finally, the graph: