Final answer:
To find the frequency of a damped oscillator, compare the damping coefficient with the critical damping coefficient to determine the damping condition, and then use the formula for damped angular frequency. Calculate the damped frequency from the angular frequency divided by 2π. Use the spring constant, mass, and damping coefficient to compute these values.
Step-by-step explanation:
To calculate the frequency of the damped oscillator, we make use of the formula for damped oscillations. However, we need to first check if the system is underdamped, critically damped, or overdamped by comparing the damping coefficient b with the critical damping coefficient bc which is defined as bc = 2√(km). In this case, the spring constant k is 2.15 × 104 N/m and the mass m is 10.7 kg, so the critical damping coefficient is bc = 2√(2.15 × 104 × 10.7). Given that the actual damping coefficient b is 4.50 N · s/m, we compare it to find the damping condition.
Assuming the system is underdamped (since the exact values of critical damping are not calculated here), the damped angular frequency ωd can be calculated with the equation ωd = √(ω02 - (β/2)2), where ω0 is the undamped angular frequency (ω0 = √(k/m)) and β is b/m. The damped frequency fd (in Hz) is then fd = ωd/(2π).
To find fd, calculate the values based on the given constants. ω0 = √(2.15 × 104 / 10.7) and β = 4.50 / 10.7. Subtracting (β/2)2 from ω0 squared gives ωd. Finally, divide ωd by 2π to find fd, the frequency of the damped oscillator.