Question

Harmonic Motion: I need help with part(b).

Answer 1

hi there

when it says compute w', i assume that you have been given the formula that transforms between w' and the w you get in **undamped** SHM, when w= sqrt (alpha).

if so: w' = w { 1 - (alpha/2w)^2 } ^ (1/2)

if instead you are required to find w' the hard way, then plug the solution they give (x = e ^ (-alpha * t /2) cos w' t )back into the original differential equation -- differentiating x twice to get x' and x'', using the product rule -- and add it all up so that it matches the original differential equation. that will allow you to compare w to w' and derive the above equation.

in terms of what it all means, critical damping is when the thing that is vibrating returns ASAP to its equilibrium position and stops there. overdamping is when it never quite gets back. under damping is when it still oscillates about the equilibrium position albeit with a gradual decline in the amplitude of oscillation.

the example often given of under damping is when the parent stops pushing the child on the swing. it is left to friction and air resistance to slow the child down, which happens over a prolonged period.

in that same example, critical damping would be the parent applying just enough *resistive* force to allow the child to stop completely when the swing next reaches its equilibrium position.