{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>
a. Find the time at which the object first passes the rest position, y = 0.

검증된 단계별 안내

Identify the function representing the displacement: y(t) = 2.5e^(-t) * cos(2t).

Set the displacement function equal to zero to find when the object first passes the rest position: 2.5e^(-t) * cos(2t) = 0.

Since 2.5e^(-t) is never zero for any real value of t, focus on solving cos(2t) = 0.

Recall that cos(θ) = 0 at odd multiples of π/2, i.e., θ = (2n+1)π/2 for n being an integer.

Set 2t = (2n+1)π/2 and solve for t to find the first positive time when the object passes the rest position. Start with n = 0 to find the smallest positive t.

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주요 개념

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

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.

Trigonometric Functions

The cosine function, represented as 'cos 2t' in the equation, is a periodic function that describes oscillatory motion. It oscillates between -1 and 1, and its argument '2t' indicates the frequency of oscillation, which affects how quickly the object moves up and down.

Finding Roots of Equations

To find when the object first passes the rest position (y = 0), we need to solve the equation '2.5e⁻ᵗ cos 2t = 0'. This involves determining the values of 't' for which the cosine function equals zero, as the exponential term is never zero. The roots of the cosine function occur at specific intervals, which can be calculated to find the desired time.