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Damped Egg on a Spring
A 50.0-{\rm g} hard-boiled egg moves on the end of a spring with force constant k = 25.0\;\rm{N/m}. It is released with an amplitude 0.300 {\rm m}. A damping force F_x = -bv acts on the egg. After it oscillates for 5.00 {\rm s}, the amplitude of the motion has decreased to 0.100 {\rm m}.
Part A  
Calculate the magnitude of the damping coefficient b.
Part A.1 How damped is it?
The system described above is ___________.
Hint A.1.a  

Hint not displayed

ANSWER:
critically damped
overdamped
underdamped
Hint A.2 What formula to use
In this problem, the motion is described by the general equation for an underdamped oscillator,

x = Ae^{-bt/2m}\cos{(\omega 't+\phi )},

where

\omega ' = \sqrt{\frac{k}{m}-\frac{b^2}{4m^2}},

x is position, and t is time. The displacement is thus a product of a oscillating cosine term and a damping term A_1(t). This equation is the solution to the damped oscillator equation m\ddot{x} = -kx - b \dot{x}.
Part A.3 Find the amplitude
What is A_1(t), the amplitude as a function of time? Use A_0 for the initial displacement of the system and m for the mass of the egg.
Hint A.3.a Initial amplitude
In the formula given in for the motion of the egg, A is the initial amplitude of the system. This means that A is a displacement at time t=0.
Give your answer in terms of A_0, m, b, and t.
ANSWER:
  A_1(t) =  A_0*e^(-b*t/(2*m)) 
Now evaluate A_1(5.00\;s) and set your answer equal to 0.100 {\rm m} as given in the problem introduction. Finally, solve for b.
Hint A.4 Solving for y in e^y

Hint not displayed

Express the magnitude of the damping coefficient numerically in kilograms per second, to three significant figures.
ANSWER:
  b =  .0220   kg/s
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