{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.

Verified step by step guidance

1. 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.
2. 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.
3. 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.
4. Solve for t: t = (2n+1)π/4. This gives the times at which the object passes the rest position.
5. 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.

Key Concepts

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.

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.

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.