an undamped mass spring system is with mass of 12kg and stiffness of 15N/mm,a)determine the natural frequency and the period of the free motion of the system b)determine the amplitude of displacement A and the phse shift for the initial conditions x0=2mm,v0=-1mm/s
To solve this problem, we'll need to use the equation for the natural frequency of a mass-spring system and the formulas for amplitude and phase shift in simple harmonic motion.
a) Determining the natural frequency and the period of the free motion:
The natural frequency of an undamped mass-spring system can be found using the equation:
ω = sqrt(k/m),
where ω is the natural frequency, k is the stiffness of the spring, and m is the mass.
First, we need to convert the stiffness from N/mm to N/m, since the mass is given in kg and the natural frequency should be in rad/s. To do this, divide the stiffness by 1000:
k = 15 N/mm = 15 N/m.
Next, we can substitute the values into the equation to find the natural frequency:
ω = sqrt(15 N/m / 12 kg) = sqrt(1.25 N/kg) ≈ 1.118 rad/s.
The period of the free motion can be calculated by using the relationship between frequency (f) and period (T):
T = 1/f.
So, the period of the free motion is:
T = 1 / 1.118 rad/s ≈ 0.895 seconds.
b) Determining the amplitude of displacement (A) and the phase shift:
To determine the amplitude and phase shift, we can use the given initial conditions and the formulas for amplitude and phase shift in simple harmonic motion:
x(t) = A * sin(ωt + φ),
where x(t) is the displacement at time t, A is the amplitude, ω is the natural frequency, and φ is the phase shift.
The amplitude of displacement (A) can be obtained from the initial displacement (x0) as:
A = |x0| = |2 mm| = 2 mm.
For the phase shift (φ), we need to use the initial velocity (v0). The phase shift can be calculated as:
φ = arctan(v0 / (ω * x0)).
Substituting the values:
φ = arctan(-1 mm/s / (1.118 rad/s * 2 mm)) ≈ -26.56 degrees.
So, the amplitude of displacement (A) is 2 mm and the phase shift (φ) is approximately -26.56 degrees.