A mass-spring-damper system oscillates with an initial amplitude of 0.2 meters and a damping coefficient of 0.1 kg/s. Determine the amplitude of the oscillations after 10 seconds.
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To determine the amplitude of the oscillations after 10 seconds, we first need to find the damping factor, which is given by the equation:
d = (b/2m)
where
d = damping factor
b = damping coefficient
m = mass of the system
Given that the damping coefficient is 0.1 kg/s, we have:
b = 0.1 kg/s
Now, let's suppose that the mass of the system is 1 kg (it's just an arbitrary choice). Therefore:
m = 1 kg
Now we can calculate the damping factor:
d = (0.1 kg/s) / (2 * 1 kg)
d = 0.05 1/s
The natural frequency of the system (ω_n) is given by the equation:
ω_n = √(k/m)
where
k = spring constant
m = mass of the system
To find the value of the spring constant, we need to use the equation of motion for the damped harmonic oscillator:
m * d²x/dt² + b * dx/dt + k * x = 0
where
m = mass of the system
b = damping coefficient
k = spring constant
x = displacement of the system
At the moment when the system is at its maximum displacement (x = 0.2 m), the only remaining force acting on the system is the one provided by the spring. Therefore, we have:
k * x = m * g
where
g = acceleration due to gravity (approximately 9.81 m/s²)
So,
k = (m * g) / x
k = (1 kg * 9.81 m/s²) / 0.2 m
k = 49.05 N/m
Now we can calculate the natural frequency:
ω_n = √(49.05 N/m / 1 kg)
ω_n = √49.05 rad/s
ω_n ≈ 7 rad/s
The amplitude of the oscillations after t seconds is given by the equation:
A = A₀ * e^(-d * t) * cos(ω_d * t)
where
A = amplitude at time t
A₀ = initial amplitude
d = damping factor
t = time
ω_d = damped angular frequency
The damped angular frequency (ω_d) is given by the equation:
ω_d = √(ω_n² - d²)
Therefore,
ω_d = √((7 rad/s)² - (0.05 1/s)²)
ω_d ≈ √(49 rad²/s² - 0.0025 1/s²)
ω_d ≈ √(48.9975 rad²/s²)
ω_d ≈ 6.994 rad/s
Plugging in the values and calculating, we have:
A = 0.2 m * e^(-0.05 1/s * 10 s) * cos(6.994 rad/s * 10 s)
A ≈ 0.2 m * e^(-0.5) * cos(69.94 rad)
Using a calculator, we find that e^(-0.5) ≈ 0.6065 and cos(69.94 rad) ≈ -0.8113. Plugging in these values:
A ≈ 0.2 m * 0.6065 * (-0.8113)
A ≈ -0.098 m
Therefore, the amplitude of the oscillations after 10 seconds is approximately 0.098 meters (or 9.8 cm).
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