A mass M is suspended from a spring and oscillates with a period of 0.940 s. Each complete oscillation results in an amplitude reduction of a factor of 0.96 due to a small velocity dependent frictional effect. Calculate the time it takes for the total energy of the oscillator to decrease to 0.50 of its initial value.

The time,t = 7.99seconds

Explanation:

Total energy of SHM system is:

E = ½kA²

0.50E = 0.50(½kAo²) = ½kA²

A = Ao/√2 = 0.707Ao

The amplitude after each oscillation is given by:

A(n) = Ao(0.96)^n

Solve for n by setting this equal to half-energy amplitude:

Ao(0.96)^n = Ao/√2

ln(Ao(0.96)^n) = ln(Ao/√2) 

ln(Ao)+n*ln(0.96) = ln(Ao)-ln(√2)

n = -ln(√2)/ln(0.96)

n = 8.5

Find time from n & period T:

Period , T = 0.940seconds

t = 8.5 × 0.940

t = nT = 7.99s

it would be c. predictability

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.