How to Calculate Amplitude Decrease in Damped Harmonic Motion After 10 Cycles?

[Tryingtobecomeanengi]

TL;DR Summary: I have to calculate the relative percentage with which the amplitude decreases after 10 cycles.

I have to calculate the relative percentage with which the amplitude decreases after 10 cycles.

When i did the calculations myself i also took the new angular frequency into account when calculating the period. I plugged this time into the following formula:

$$x(t) = A e^{-\frac{b}{2m} t}$$

But when i looked at the solution, they did not use this new angular frequency, but the original one to calculate the time. Is the change in frequency already taken into account in the dampening constant, or did i miss something?

This is the exact question:

A mass of 0.500kg is attached to a spring with a spring constant of 50.0N/m. At time t = 0, the
mass reaches its maximal velocity of 20.0m/s and moves to the left.
1. What is the frequency of oscillation?
2. Determine the equation of motion of the mass, i.e. its position as a function
of time. The equilibrium position is set at x = 0.
3. At which distance from the equilibrium position is the potential energy of
the system 3 times higher than the kinetic energy?
4. Determine the largest time in the cycle it takes for the mass to cover a total
distance of 1 meter?
5. What is the relative percentage of the amplitude decrease after 10 cycles of
the motion if a damper is added to the system? The value of the damping
constant is 2 kg/s.

 

[kuruman]

But when i looked at the solution, they did not use this new angular frequency, but the original one to calculate the time.

The only time you are asked to find is in part 4. This is before the damper is added in part 5.

 

[tech99]

As far as I know a damped waveform does not exhibit a change in frequency. It does contain additional frequency components which arise due to (a) the distorted shape of the damped waves and (b) the sudden commencement of oscillation.

 

[TSny]

But when i looked at the solution, they did not use this new angular frequency, but the original one to calculate the time.

I think they should have used the damped frequency instead of the undamped frequency. I find that these two frequencies differ only by about 2%. But I get that this makes about a 12% difference in the result for part 5.