38 lines
852 B
TeX
38 lines
852 B
TeX
\documentclass{article}
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\usepackage{amsmath}
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\begin{document}
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The velocity of the mass as a function of time \( t \) is described by the equation:
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\[
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v(t) = v_{\text{max}} \cdot \sin(\omega t + \phi)
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\]
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where:
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\begin{itemize}
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\item \( v_{\text{max}} \) is the maximum velocity (amplitude of velocity),
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\item \( \omega \) is the angular frequency of the oscillations,
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\item \( \phi \) is the initial phase.
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\end{itemize}
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The angular frequency \( \omega \) is related to the period \( T \) by the formula:
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\[
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\omega = \frac{2\pi}{T}
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\]
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For a period \( T = 0.4 \) seconds:
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\[
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\omega = \frac{2\pi}{0.4} = 5\pi \, \text{rad/s}
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\]
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If the mass starts moving from the equilibrium position with an upward velocity, the initial phase \( \phi \) is 0. Then, the velocity equation becomes:
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\[
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v(t) = 0.8 \cdot \sin(5\pi t)
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\]
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\end{document}
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