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: