Final answer:
To find the mass of the fish, the spring constant was first calculated using Hooke's Law, resulting in 1434.78 N/m. Then, the mass was determined using the formula for frequency of a mass-spring system in simple harmonic motion, yielding a mass of 36.34 kg for the fish.
Step-by-step explanation:
To determine the mass of the fish that is oscillating at the end of the spring, we first need to calculate the spring constant (k) using Hooke's Law, which states that the force (F) exerted by a spring is equal to the spring constant (k) multiplied by the displacement (x) from its equilibrium position:
F = kx
With a maximum weight (force) of 165 N and a displacement at that weight of 11.5 cm (0.115 m), the spring constant can be calculated as:
k = F / x = 165 N / 0.115 m = 1434.78 N/m
Now, knowing that the spring is oscillating with a frequency (f) of 2.80 Hz, we can use the formula that relates the frequency, spring constant, and mass (m) in a mass-spring system undergoing simple harmonic motion:
f = (1/2π) * √(k/m)
Solving for m, we get:
m = k / (4π²f²)
Substituting the values:
m = 1434.78 N/m / (4π² * (2.80 Hz)²) = 1434.78 N/m / (4π² * 7.84 Hz²) = 1434.78 / (39.4784) = 36.34 kg
Therefore, the mass of the fish is 36.34 kg.