A mass of 0.394 kg is attached to a spring with a spring constant of 224.2 N/m. Its oscillation is damped, with damping constant b = 14.0 kg/s. What is the angular frequency of this damped oscillation?
ω = rad/s
To find the angular frequency (ω) of a damped oscillation, we can use the formula:
ω = √(k/m - (b/2m)²)
where:
- k is the spring constant (224.2 N/m)
- m is the mass attached to the spring (0.394 kg)
- b is the damping constant (14.0 kg/s)
Let's plug in the given values into the formula to find the angular frequency:
ω = √(224.2 N/m / 0.394 kg - (14.0 kg/s / (2 * 0.394 kg))²)
First, let's simplify the expression inside the square root:
ω = √(569.54 N/kg - (17.77 kg/s)²)
ω = √(569.54 N/kg - 315.12 kg²/s²)
Next, let's subtract the values inside the square root:
ω = √(254.42 N/kg - 315.12 kg²/s²)
Now, let's simplify the expression inside the square root further:
ω = √(-60.7 kg²/s²)
Since the expression inside the square root is negative, it means that the given system is over-damped, where the damping is greater than the critical damping. In this case, the angular frequency is purely imaginary.
Therefore, the angular frequency (ω) of this damped oscillation is not a real number but is instead imaginary.