An unforced, under damped harmonic oscillator moves with velocity Vsub(0) throug

Question

An unforced, under damped harmonic oscillator moves with velocity  Vsub(0) through position Xsub(0) at time t=0. Find it's displacement as a function of time, that is, use these initial conditions to solve for the arbitrary constants in the expression for x=x(t). There is no need to derive the form of the solution for x(t), just use the solution to find the two arbitrary constants in terms of Vsub(0), Xsub(0), and gamma (the dampening parameter) and the oscillator frequency omega.

use the following equation: where X is equal to the following:

An unforced, under damped harmonic oscillator moves with velocity Vsub(0) through position Xsub(0) at time t=0. Find it's displacement as a function of time, that is, use these initial conditions to solve for the arbitrary constants in the expression for x=x(t). There is no need to derive the form of the solution for x(t), just use the solution to find the two arbitrary constants in terms of Vsub(0), Xsub(0), and gamma (the dampening parameter) and the oscillator frequency omega. use the following equation: where X is equal to the following:= ae-gammat cos(omegat-theta),

Explanation / Answer

x(0) = a cos(theta)

so a = x0/cos(theta)

v = x' = - a gamma e^(-gamma t) cos (wt - theta) - a e^(-gamma t) w sin (wt - theta)

v(0) = - a gamma cos(theta) - a w sin(theta) = v0

- gamma x0 - x0 w tan(theta) = v0

-x0 w tan theta = v0 + gamma x0

tan theta = (- v0 - gamma x0)/( x0 w)

so a = x0/ cos(arctan( (- v0 - gamma x0)/( x0 w))

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.