a)
[m]
b) 0.033 s
c) -0.152 m/s
Step-by-step explanation:
a)
The force acting on the mass-spring system is (restoring force)

where
k = 9 is the spring constant
y is the displacement
Also, from Newton's second law of motion, we know that

where
m = 1 g = 0.001 kg is the mass
y'' is the acceleration
Combining the two equations,

This is a second order differential equation; the solution for y(t) is

where
A is the amplitude of motion
is the angular frequency
The spring starts its motion from its equilibrium position, this means that y=0 when t=0; therefore, the phase shift must be

So the displacement is

The velocity of the spring is equal to the derivative of the displacement:

We know that at t = 0, the initial velocity is 6 in/s; since 1 in = 2.54 cm = 0.0254 m,

And since at t = 0, 
Then we have:

From which we find the amplitude:

So the solution for the displacement is
[m]
b)
Here we want to find the time t at which the mass returns to equilibrium, so the time t at which

This means that

We know already that the first time at which this occurs is
t = 0
Which is the beginning of the motion.
The next occurence of y = 0 is instead when

which means:

c)
As said in part a), the velocity of the mass-spring system at time t is given by the derivative of the displacement, so

where we have
is the angular frequency
is the amplitude of motion
t is the time
Here we want to find the velocity of the mass when the time is that calculated in part b):
t = 0.033 s
Substituting into the equation, we find:
