A spring is attached to the ceiling and pulled 8 cm down from equilibrium and released. The amplitude decreases by 17% each second. The spring oscillates 19 times each second. Find an equation for the distance, D the end of the spring is below equilibrium in terms of seconds, t.

D(t) = 8(0.83)^(t) cos 38πt

Explanation:

We are told that the spring oscillates 19 times each second.

Thus, period = 1/19

We are also told that it's pulled 8cm downwards and the amplitude decreases by 17% each second.

Thus;

Amplitude;A = 8 × (1 - (17/100))^(t)

A = 8(0.83)^(t)

If we consider the function;

y = A cos (bx - c) + d

Now, 2π/b = period

So, 1/19 = 2π/b

b = 38π

So, D(t) = 8(0.83)^(t) cos (38πt - c) + d

Since we started from minimum,

Vertical shift, d = 0 and horizontal shift c = 0

So,we now have;

D(t) = 8(0.83)^(t) cos 38πt

russian is not my first language, so i'm afraid i can't write out an answer in russian, but hopefully you can find a way to do so. here's my best translation of the problem: you're given a plot of a particle's position in one-dimensional space over time, and you're asked to find either the total distance traveled by the particle after 6 seconds had elapsed, or the total displacement of the particle after 6 seconds.

if you want the total distance traveled, you would first need to determine the particle's velocity as a function of time, then integrate that over the 6-second interval. we know the position function is quadratic with roots at 0 and 6, so we have

where is unknown. because the position is parabolic, there is symmetry about the midpoint at , and we know that

so that the position function is

which means the velocity is

then the total distance traveled is

so the total distance traveled is 20 m.

on the other hand, if you want to find displacement: the particle starts at and ends at , so its displacement is 0.