(a) 0.5 s
In a simple harmonic motion, the period is equal to the reciprocal of the frequency:
where f is the frequency
For this simple harmonic oscillator, the frequency is
f = 2.0 Hz
So the period is
(b) 12.56 rad/s
The angular frequency is given by
where
f is the frequency
In this problem,
f = 2.0 Hz
So the angular frequency is
(d) 0.419 rad
The displacement of the system can be written as
(1)
where A is the amplitude and is the phase constant.
The velocity is the derivative of the displacement:
(2)
Here we know that
at t=0, x(5)=5.0 cm and v(t)=-28 cm/s. So we can rewrite the ratio (2)/(1) as
And re-arranging the equation we can find the phase constant:
(c) 5.47 cm
The displacement of the system can be written as
(1)
at t=0, x=5.0 cm, so using the values we found for we can now solve the equation to find A, the amplitude:
(e) 68.7 cm/s
The maximum speed in a simple harmonic system is given by
where in this case we have
Substituting the numbers into the formula, we find
(f) 862.9 cm/s^2
The maximum acceleration in a simple harmonic system is given by
where in this case we have
Substituting the numbers into the formula, we find
(g) 0.04 J
The total energy of the system is equal to the kinetic energy when the speed of the system is maximum: this occurs at x=0 (equilibrium position), where the elastic potential energy is zero, and all the energy is just kinetic energy:
where we have
m = 170 g = 0.170 kg is the mass
is the maximum speed
Substituting into the equation,
(h) 3.65 cm
The position of the system is given by
where we have
is the angular frequency
is the amplitude
is the phase constant
Substituting t=0.4 s, we find the position at this time: