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A class looked at the oscillation of a mass on a spring. They observed for 10 seconds and found its oscillation was well fit by assuming the mass's
motion was governed by Newton's second law with the static-spring force *F*^{s-s} = -*k* Δ*s*
where Δ*s* represents the stretch or squeeze of the spring and *k* is the spring constant. However, it was also
clear that this was not an adequate representation for times on the order of 10 minutes, since by that time the mass had stopped oscillating.

- Suppose the mass was started at an initial position
*x*_{0}with a velocity 0. Write down the solution,*x*(*t*) and*v*(*t*), for the equations of motion of the mass using only the static-spring force. What is the total energy of the oscillating mass? - Assume that there is also a velocity dependent force (the
*damping*force) that the spring exerts on an object when it is moving. Let's make the simplest assumption, that the dynamic-spring force is linear in the velocity,*F*^{d-s}= -γ*v*. Assume further that over one period of oscillation the velocity dependent piece is small. Therefore, take*x*(*t*) and*v*(*t*) to be given by the oscillation without damping. Calculate the work done by the damping force over one period in terms of the parameters*k*,*m*, and γ and the fraction of the energy lost in one period. - The mass used was 1 kg. From the description of the experiment, estimate the spring constant and the number of periods it took to lose half the energy. Use this and your result from (a) to estimate the approximate size of the damping constant γ.

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Page last modified October 30, 2002: O15