At what distance from equilibrium is the energy of a sho half kinetic and half potential?

Distance from equilibrium x = 0.707A

Where A is the amplitude of the SHO

Explanation:

all the energy of a SHO is a potential energy at

PE = 0.5kA^2

Where A is the amplitude of the SHO

Half that energy means

PE = 0.25kA^2

The energy of a SHO is half KE and half PE at a displacement of

PE = 0.5kx^2

Where x is the displacement the displacement from equilibrium when the energy is half potential energy and half kinetic energy.

Equating both equations and solving, we have,

0.25kA^2 = 0.5kx^2

We solve for x

x^2 = (0.25kA^2)/(0.5kx^2)

x^2 = 0.5A^2

x = 0.707A.

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.