Can someone please show me how to work out this problem, the answer was given as 206 ms
1. If the damping coefficient in the damped harmonic oscillator is 11.2 kg/s and m = 0.5 kg, k = 100N/m, how long to the nearest millisecond will it take the amplitude of the oscillator to decay to 0.1 of its initial value?
1. If the damping coefficient in the damped harmonic oscillator is 11.2 kg/s and m = 0.5 kg, k = 100N/m, how long to the nearest millisecond will it take the amplitude of the oscillator to decay to 0.1 of its initial value?
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The term 'damping coefficient' has 2 common meanings:
1)It is the constant c in the equation md²x/dt² + cdx/dt + kx = 0 (damped SHM). Sometimes 'b' is used.
2)It is γ (gamma), where γ = c/2m (or b/2m).
From the unit you give (kg/s), it appears to be the 1st meaning.
If you look through the 1st link you'll see what I mean.
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The value of k doesn't affect the solution to this problem.
The amplitude decays according to A = Aoe^(-γt) E.g. look through the 2nd link.
where
A(t) = amplitude at time t
Ao = amplitude at time 0
γ = damping coefficient (2nd meaning)
γ = c/2m = 11.2/(2x0.5) = 11.2s⁻¹
If amplitude decays to 0.1 of its initial value, A(t) = 0.1Ao
A(t) = Aoe^(-γt)
0.1Ao = Aoe^(-γt)
0.1= e^(-γt)
Take natural logarithms of both sides
ln(0.1) = -γt
-2.303 = -11.2t
t = 0.2056s
(=206ms to the nearest ms)
1)It is the constant c in the equation md²x/dt² + cdx/dt + kx = 0 (damped SHM). Sometimes 'b' is used.
2)It is γ (gamma), where γ = c/2m (or b/2m).
From the unit you give (kg/s), it appears to be the 1st meaning.
If you look through the 1st link you'll see what I mean.
________________
The value of k doesn't affect the solution to this problem.
The amplitude decays according to A = Aoe^(-γt) E.g. look through the 2nd link.
where
A(t) = amplitude at time t
Ao = amplitude at time 0
γ = damping coefficient (2nd meaning)
γ = c/2m = 11.2/(2x0.5) = 11.2s⁻¹
If amplitude decays to 0.1 of its initial value, A(t) = 0.1Ao
A(t) = Aoe^(-γt)
0.1Ao = Aoe^(-γt)
0.1= e^(-γt)
Take natural logarithms of both sides
ln(0.1) = -γt
-2.303 = -11.2t
t = 0.2056s
(=206ms to the nearest ms)