Avery light cotton tape is wrapped around the outside surface of a uniform cylinder of mass m and radius r. the free end of the tape is attached to the ceiling. the cylinder is released form rest and as it descends it unravels from the tape without slipping. the moment of inertia of the cylinder about its center is i = 1/2 mr2
a. on the diagram above show all the forces applied on the cylinder.
b. find the acceleration of the center of the cylinder when it moves down.
c. find the tension force in the tape.

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.

find the sum of the two horizontal vectors and the sum of the two vertical vectors

isoceles right angled triangle