Oscillations in mechanical structures and electric circuits can often be described by the function

where t is time and ( is the oscillation frequency in radians per unit time. The oscillations have a period of 2π/ω, and their amplitudes decay in time at a rate determined by τ, which is called the time constant. The smaller τ is, the faster the oscillations die out.

a. Use these facts to develop a criterion for choosing the spacing of the t values and the upper limit on t to obtain an accurate plot of y(t). Consider two cases: 4τ> 2π/ω and 4τ < 2π/ω.

b. Apply your criterion, and plot y(t) for τ = 10, ω = π, and ϕ = 2.

c. Apply your criterion, and plot y(t) for τ = 0.1, ω = 8π, and ϕ = 2.

y(t) = e sin(ot + 4) SI