A book of mass 7 kg rests on a plank. You tilt one end of the plank and slowly increase the angle of the tilt. The coefficient of static friction between the book and the plank is 0.32. What is the maximum angle of tilt for which the book will remain stationary and not slide down the plank

To solve this problem we will apply the concepts related to the Friction force and the force induced by gravity. Since the displacement is in angular mode, the component of the horizontal force of friction will be equivalent to the component of the vertical force of gravity. For balance to exist, both must be equal

Where,

m = Mass

g = Gravitational acceleration

= Constant of friction

Then,

Therefore the maximum angle  of tilt for which the book will remain stationary and not slide down the plank is 17.74°

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.