Question

X(t)=x₀e− γ/²ᵗ cos(ωt+ϕ)

Plot this solution for 0≤t≤6, with the parameters x 0=1, ω=2π, and ϕ=−π/2, for three different damping rates γ=0,γ=0.5 and γ=3. Plot all the curves on the same axes in different colours so they can be compared.

Answer 1

To plot x(t) for different damping rates, substitute the given values and calculate the positions for each time interval.

Step-by-step explanation:

To plot the function x(t) = x₀e- γ/²ᵗ cos(ωt+ϕ) for different damping rates, we can use the given parameters. Let's calculate the values of x(t) for each damping rate and plot them on the same axes:

For γ = 0, the function becomes x(t) = x₀ cos(ωt+ϕ). This is a simple harmonic motion without any damping, resulting in a cosine function. We can plot this using the given values of x₀ = 1, ω = 2π, and ϕ = -π/2.

For γ = 0.5 and γ = 3, we use the formula x(t) = x₀e- γ/²ᵗ cos(ωt+ϕ) with the same given values. We can calculate the values of x(t) for each time interval and plot them on the same axes, using different colors for each damping rate.