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## Homework Statement

This problem is taken from Problem 2.3, Introduction to Vibration and Waves, by H.J. Pain and P. Rankin:

A critically mechanical system consisting of a pan hanging from a spring with a damping. What is the value of damping force r if a mass extends the spring by 10cm without overshoot. The mass is 5kg. (g=9.81ms^(-2)).

## Homework Equations

$$m\ddot{x} = F_{spring}+F_{damping}+F_{gravity} = -sx -r\dot{x} + mg$$

## The Attempt at a Solution

Using second Newton law, I can write the equation of the system:

$$m\ddot{x} = F_{spring}+F_{damping}+F_{gravity} = -sx -r\dot{x} + mg$$. Rewriting the equation, we obtain:

$$ 5\ddot{x} + r \dot{x} + s x = 5g$$

Because the system is critically damped, we have r

^{2}- 4ms = 0 so we can remove the s from the equation (s=r

^{2}/(4m) = r

^{2}/20):

$$ \ddot{x} + \frac{r}{5} \dot{x} + \frac{r^2}{4} x = g$$

Supposing what I've written above is correct, I can obtain the equation of motion. The homogeneous solution is

$$ x_h(t) = e^{-\frac{r}{2m}t} (A + Bt) = e^{-\frac{r}{10}t} (A+Bt)$$

where A, B constant determined by the initial conditions. The particular solution is:

$$ x_p(t) = \frac{4g}{r^2} $$

So the complete equation of motion is:

$$ x(t) = \frac{4g}{r^2} + e^{-\frac{r}{2m}t} (A + Bt) $$

Actually I have two questions:

- What are the initial conditions? I have supposed $$x(0)=0, \dot{x}(0)=0$$ but this leads to A = B = 0;

- Where do I have to plug the Δx = 10cm data in order to obtain r? Is it possible I need to use the energy formula?

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