Damped mass spring system oscillates at 205 Hz. The time constant of the system is 6.6s. At t=0 the amplitude of oscillation is 4.7 cm and the energy of the oscillating system is 26J. What is the amplitude of oscillation at t=9.9s? Answer in units of cm
please help
y = .047 e^-(t/6.6)cos (410 pi t)
But all we need is e^ -(9.9/6.6)
= .223
and
.223*4.7 = 1.05 cm
thank you very much
To find the amplitude of oscillation at t=9.9s, we need to understand the behavior of a damped mass spring system.
The general equation for the displacement of a damped mass spring system is given by:
x(t) = Ae^(-bt)cos(ωt + φ)
Where:
- x(t) represents the displacement of the mass from its equilibrium position at time t.
- A is the initial amplitude of oscillation.
- b is the damping coefficient, related to the time constant by b = 1/τ.
- ω represents the angular frequency of oscillation, given by ω = 2πf, where f is the frequency of oscillation.
- φ is the phase angle.
In this case, we are given the frequency of oscillation f = 205 Hz, and the time constant τ = 6.6s.
First, we can calculate the angular frequency ω:
ω = 2πf = 2π * 205 Hz ≈ 1287.78 rad/s
Next, we can calculate the damping coefficient b:
b = 1/τ ≈ 1/6.6s ≈ 0.1515 rad/s
Given that the amplitude of oscillation at t=0 is 4.7 cm, we can substitute these values into the equation to find the initial displacement:
x(0) = Ae^(-bt)cos(ωt + φ)
4.7 cm = A * e^(-0 * 0) * cos(ω * 0 + φ)
4.7 cm = A * cos(φ)
Since the initial displacement is 4.7 cm, we have:
4.7 cm = A * cos(φ) ... (Equation 1)
To find the amplitude at t=9.9s, we can substitute t = 9.9s into the equation and solve for the new amplitude:
x(9.9) = Ae^(-b * 9.9) * cos(ω * 9.9 + φ)
To simplify the equation, we divide both sides by Ae^(-b * 9.9):
cos(ω * 9.9 + φ) = x(9.9) / (Ae^(-b * 9.9))
Now, we can substitute the given values:
cos(ω * 9.9 + φ) = x(9.9) / (4.7 cm * e^(-0.1515 rad/s * 9.9))
To find the new amplitude, we multiply both sides by A and use Equation 1 to eliminate φ:
A * cos(ω * 9.9 + φ) = x(9.9) / (4.7 cm * e^(-0.1515 rad/s * 9.9))
A * (x(9.9) / (4.7 cm * e^(-0.1515 rad/s * 9.9))) = A * cos(φ)
Finally, we solve for the new amplitude at t=9.9s:
New Amplitude = x(9.9) / (e^(-0.1515 rad/s * 9.9))
Substituting the given values into the equation, we can calculate the new amplitude.