(a) 3539 N/m
Hook's law states that:
where F is the force applied on the spring, k is the spring constant, x is the stretching of the spring.
In this problem, we have:
is the force applied (the weight of the fish)
is the stretching of the spring
Solving the equation for k, we find the spring constant:
(b) 0.85 s
The period of oscillation of a spring-mass system is
In this case,
m = 65.0 kg
k = 3539 N/m
Substituting into the formula,
(c) 0.37 m/s
The initial elastic potential energy of the spring when the fish is pulled down is:
where
is the stretching of the spring with respect to the initial position
Substituting,
The spring reaches its maximum speed when it crosses the equilibrium position, for which , so when all the elastic potential energy has been converted into kinetic energy:
where v is the speed of the fish. Solving for v, we find