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- Homework Statement
- A swingboat with mass m = 130 kg is approximately taken to be a mathematical pendulum with a suspension length l = 3.5 m.

The swingboat undergoes a damped harmonic oscillation after the motor is turned off (at t = 0) with the form Φ(t) = Φ_0 * exp(-γt)*cos(ωt) with a starting amplitude Φ(t=0) = 15° and a frequency of f = 0.27 Hz. The amplitude is reduced to 12° after 5 oscillation periods.

a) Calculate the damping coefficient γ

b) How long does it take until the amplitude reaches 2° ?

- Relevant Equations
- T = 1/f

Hi,

so of course Φ

I need help with b).

If I do 2° = 15° * exp(-0.012t)*cos(2πf*t), I'm not able to find t so I did something else by assuming that the amplitude decreases at a constant rate:

After 5*T = 5*1/f = 18.52 s, the amplitude decreases by 3°.

So I have 18.52 s --> ΔΦ = 3°.

Which means after 1 s the amplitude decreases by 3°/18.52 = 0.162°

15° - 0.162°*x = 2° ⇔ x = 80.25

Which means it would take 80.25 s. But if I plug in t = 80.25 s into Φ(t) = Φ_0 * exp(-γt)*cos(ωt) I don't get 2° so my method was wrong.

I would like to ask why my method is wrong and how I could solve this problem. How could I solve the equation 2° = 15° * exp(-0.012t)*cos(2πf*t)? I thought I could use Euler's formula for cos(2πf*t) so that I could write the right hand side of the equation as a single e to the power of something, but it gets me nowhere.

so of course Φ

_{0}= 15° and after solving after solving Φ(t=5*T = 5/f) I found γ = 0.012I need help with b).

If I do 2° = 15° * exp(-0.012t)*cos(2πf*t), I'm not able to find t so I did something else by assuming that the amplitude decreases at a constant rate:

After 5*T = 5*1/f = 18.52 s, the amplitude decreases by 3°.

So I have 18.52 s --> ΔΦ = 3°.

Which means after 1 s the amplitude decreases by 3°/18.52 = 0.162°

15° - 0.162°*x = 2° ⇔ x = 80.25

Which means it would take 80.25 s. But if I plug in t = 80.25 s into Φ(t) = Φ_0 * exp(-γt)*cos(ωt) I don't get 2° so my method was wrong.

I would like to ask why my method is wrong and how I could solve this problem. How could I solve the equation 2° = 15° * exp(-0.012t)*cos(2πf*t)? I thought I could use Euler's formula for cos(2πf*t) so that I could write the right hand side of the equation as a single e to the power of something, but it gets me nowhere.