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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblema 4.8.37c
Capitolo 4, Problema 4.8.37c

{Use of Tech} A damped oscillator The displacement of an object as it bounces vertically up and down on a spring is given by y(t) = 2.5e⁻ᵗ cos 2t, where the initial displacement is y(0) = 2.5 and y = 0 corresponds to the rest position (see figure). <IMAGE>


c. Find the time at which the object passes the rest position for the second time.

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First, understand the function y(t) = 2.5e⁻ᵗ cos 2t. This represents a damped oscillator, where the term 2.5e⁻ᵗ is the damping factor and cos 2t is the oscillatory component.
To find when the object passes the rest position, set y(t) = 0 and solve for t. This means solving the equation 2.5e⁻ᵗ cos 2t = 0.
Since 2.5e⁻ᵗ is never zero for any real value of t, focus on the oscillatory part: cos 2t = 0. The cosine function is zero at odd multiples of π/2, so set 2t = (2n+1)π/2, where n is an integer.
Solve for t: t = (2n+1)π/4. This gives the times at which the object passes the rest position.
To find the second time the object passes the rest position, use n = 1 (since n = 0 corresponds to the first time). Substitute n = 1 into the equation t = (2n+1)π/4 to find the second time.

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Damped Oscillator

A damped oscillator is a system in which the amplitude of oscillation decreases over time due to energy loss, often modeled by an exponential decay function. In the given equation, the term '2.5e⁻ᵗ' represents this damping effect, indicating that the displacement diminishes as time progresses.
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Cosine Function in Oscillations

The cosine function is fundamental in describing periodic motion, such as oscillations. In the equation y(t) = 2.5e⁻ᵗ cos 2t, the 'cos 2t' component indicates the oscillatory behavior of the system, with '2' representing the angular frequency, which affects the speed of oscillation.
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Finding Roots of the Function

To determine when the object passes the rest position (y = 0), we need to find the roots of the function y(t). This involves solving the equation 2.5e⁻ᵗ cos 2t = 0, which occurs when cos 2t = 0, leading to specific values of t that correspond to the times the object crosses the rest position.
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