Exponential Damping 5

Exponential Damping 5#

A params.m kg mass oscillates on a params.k N/m spring. The damping constant of this spring is b = params.b kg/s.

Useful Info#

For slowly moving objects we’ve seen that the drag force grows in proportion to the velocity, D→=−bv→, where b is the damping constant and v→ is the velocity of the object.

The net force acting on a slowly moving mass attached to a massless spring in the presence of a drag force (for motion along x relative to an equilibrium point x0) can be written as:

(14)#F_net,x=−bdxdt−kx=ma=md2xdt2

The solution of this differential equation is found to be \begin{equation} x(t) = Ae^{-\frac{bt}{2m}} \cos(\omega t) = Ae^{-\frac{t}{2\tau}} \cos(\omega t)= A(t) \cos(\omega t) \end{equation} , where A is the initial amplitude of the oscillation, τ is the time constant, A(t) is the time-dependent amplitude of the oscillation, and ω=km−b24m2 is the angular frequency of the damped oscillation.

Part 1#

What is the amplitude of the spring divided by the amplitude at t=0 s (A) after one period of oscillation? Answer the question to two significant digits.

Answer Section#

Please enter in a numeric value in .

Attribution#

Problem is licensed under the CC-BY-NC-SA 4.0 license.
The Creative Commons 4.0 license requiring attribution-BY, non-commercial-NC, and share-alike-SA license.