A 50-cm-long spring is suspended from the ceiling. A 410 g mass is connected to the end and held at rest with the spring unstretched. The mass is released and falls, stretching the spring by 16 cm before coming to rest at its lowest point. It then continues to oscillate vertically. Part A What is the spring constant

25.125 N/m

Explanation:

extension on the spring e = 16 cm 0.16 m

mass of hung mass m = 410 g = 0.41 kg

equation for the relationship between force and extension is given by

F = ke

where k is the spring constant

F = force = mg

where m is the hung mass,

and  g is acceleration due to gravity = 9.81 m/s^2

imputing value, we have

0.41 x 9.81 = k x 0.16 = 0.16k

4.02 = 0.16k

spring constant k = 4.02/0.16 = 25.125 N/m

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.