The effective spring constant is 6391.3 N/m and the frequency of the oscillations is approximately 0.569 Hz.
To calculate the effective spring constant and frequency of the oscillations, we can use the equation for the period of a mass-spring system:
T = 2π√(m/k)
Step 1: Calculate the effective spring constant (k)
Using the given information, we have:
Initial mass of the boat = 175 kg
Mass of the person = 75.0 kg
Change in displacement = 11.5 cm = 0.115 m
Weight of the person = Mass of the person × Acceleration due to gravity = 75.0 kg × 9.8 m/s² = 735 N
From Hooke's Law, we know that the force exerted by the spring is equal to the weight of the person when displaced:
F = kx, where F is the force, k is the spring constant, and x is the displacement
Thus, k = F/x = 735 N / 0.115 m = 6391.3 N/m
Step 2: Calculate the frequency (f)
Using the formula T = 1/f, where T is the period and f is the frequency, we can rearrange the formula for the period:
T = 2π√(m/k)
1/f = 2π√(m/k)
f = 1/(2π)√(k/m)
Substituting the values, we get:
f = 1/(2π)√(6391.3 N/m / (175 kg + 75.0 kg)) = 1/(2π)√(6391.3 N/m / 250 kg) ≈ 0.569 Hz
Therefore, the effective spring constant is 6391.3 N/m and the frequency of the oscillations is approximately 0.569 Hz.