75. {Use of Tech} Oscillator displacements Suppose a mass on a spring that is slowed by friction has the position function:
s(t) = e⁻ᵗ sin t
a. Graph the position function. At what times does the oscillator pass through the position s = 0?

검증된 단계별 안내

Identify the position function given: \(s(t) = e^{-t} \sin t\). This function describes the displacement of the mass on the spring over time.

To find when the oscillator passes through the position \(s = 0\), set the position function equal to zero: \(e^{-t} \sin t = 0\).

Since \(e^{-t}\) is never zero for any real \(t\), the zeros of \(s(t)\) occur when \(\sin t = 0\).

Recall that \(\sin t = 0\) at integer multiples of \(\pi\), so the times when the oscillator passes through \(s=0\) are \(t = n\pi\), where \(n\) is any integer (0, 1, 2, 3, ...).

To graph the position function, plot \(s(t) = e^{-t} \sin t\) over a suitable interval (for example, \(t \geq 0\)), noting that the amplitude of the oscillations decreases exponentially due to the \(e^{-t}\) factor, and the zeros occur at multiples of \(\pi\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Damped Harmonic Motion

Damped harmonic motion describes oscillations where the amplitude decreases over time due to friction or resistance. In the given function s(t) = e⁻ᵗ sin t, the exponential term e⁻ᵗ represents the damping effect, causing the oscillations to gradually reduce in magnitude as time increases.

Zeros of a Function

Finding when the oscillator passes through position s = 0 involves solving s(t) = 0. Since s(t) is a product of e⁻ᵗ and sin t, and e⁻ᵗ is never zero, the zeros occur when sin t = 0. This happens at integer multiples of π, i.e., t = nπ for n = 0, 1, 2, ...

Graphing Transcendental Functions

Graphing s(t) = e⁻ᵗ sin t requires understanding how the sinusoidal oscillations are modulated by the exponential decay. The graph shows oscillations with peaks decreasing exponentially, illustrating the combined effect of sine wave behavior and damping over time.

75. {Use of Tech} Oscillator displacements Suppose a mass on a spring that is slowed by friction has the position function:s(t) = e⁻ᵗ sin ta. Graph the position function. At what times does the oscillator pass through the position s = 0?