6. Consider a harmonic oscillator with spring constant k and mass m placed in a

Question

6. Consider a harmonic oscillator with spring constant k and mass m placed in a critically damped medium. It has an initial position xo and is given an initial velocity vo (a) Using the general solution derived in class, find x(t) and v(t) in terms of the above given quantities. (b) If xo is positive and vo negative (to the left), find the minimum speed lvol required to make the mass overshoot the equilibrium position. (c) For this part assume instead that vo 0, and the particle is released from rest atxo. Find the total energy dissipated by friction as the particle comes to rest at its equilibrium position. [Hint: This is the dissipated power (damping force x velocity) integrated over all time.] Show that the total dissipated energy is exactly equal to the initial potential energy, and discuss why this makes sense. You might want to look up "Gamma function" or "factorial integral".

Explanation / Answer

6. given harmonic oscillator

mass = m

spring constant = k

critically damped medium

let the harmonic osscilator rquatyio be

mx" + cx' + kx = 0

from critical damped motion

c = 2*sqrt(mk)

a. we consider the roots to be x = e^lambda*t

then from critical damping

lambda = -c/2m

hence

x = e^(-2*sqrt(mk)/2m)t = Ae^(-t*sqrt(k/m))

b. xo = Ao

xo > 0

x = xo*e^(-t*sqrt(k/m))

for minimum speed to overshoot the equilibrium poisition = vo

x' = -xo*(sqrt(k/m)*e^(-t*sqrt(k/m)))

hence

vo > xo*sqrt(k/m)

c. consdidering vo = 0

xo = xo > 0

x = xo*e^(-t*sqrt(k/m))

energy dissipated by friciton = E

dE = -cx'*dx = xo*ke^(-t*sqrt(k/m)*dx

dE = k*x*dx

integrating from x = xo to x = x

E = k(xo^2 - x^2)/2

for x = 0

E = kxo^2/2

inteiial PE = kxo^2/2

hence this is the intiial PE, and all this energy has been dissipated when the pendulum comes to erst finally hence following conservation fo enregy

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