16.7 Damped harmonic motion  (Page 3/6)

1. By equating the work done to the energy removed, solve for the distance $d$ .
2. The work done by the non-conservative forces equals the initial, stored elastic potential energy. Identify the correct equation to use:
   ${\text{W}}_{\text{nc}}=\Delta \left(\text{KE}+\text{PE}\right)={\text{PE}}_{\text{el,f}}-{\text{PE}}_{\text{el,i}}=\frac{1}{2}k\left({\left(\frac{{\mu }_{\mathrm{k}}\mathit{\text{mg}}}{k}\right)}^2-X^2\right).$
3. Recall that $W_{\text{nc}}=–\text{fd}$ .
4. Enter the friction as $f={\mu }_{\text{k}}\text{mg}$ into $W_{\text{nc}}=–\text{fd}$ , thus
   $W_{\text{nc}}={–\mu }_{\text{k}}\text{mgd}.$
5. Combine these two equations to find
   $\frac{1}{2}k\left({\left(\frac{{\mu }_k\text{mg}}{k}\right)}^2-X^2\right)=-{\mu }_{\text{k}}\text{mgd}.$
6. Solve the equation for $d$ :
   $d=\frac{\text{k}}{{\text{2}\mu }_{\text{k}}\text{mg}}(X^2–{\left(\frac{{\mu }_{\text{k}}\text{mg}}{k}\right)}^2).$
7. Enter the known values into the resulting equation:
   $d=\frac{\text{50}\text{.}\mathrm{0\; N/m}}{2\left(0\text{.}\text{0800}\right)\left(0\text{.}\text{200}\text{kg}\right)\left(9\text{.}\text{80}{\text{m/s}}^2\right)}\left({\left(0\text{.}\text{100}\text{m}\right)}^2-{\left(\frac{\left(0\text{.}\text{0800}\right)\left(0\text{.}\text{200}\text{kg}\right)\left(9\text{.}\text{80}{\text{m/s}}^2\right)}{\text{50}\text{.}0\text{N/m}}\right)}^2\right).$
8. Calculate $d$ and convert units:
   $d=1\text{.}\text{59}\text{m}.$

Discussion b

This is the total distance traveled back and forth across $x=0$ , which is the undamped equilibrium position. The number of oscillations about the equilibrium position will be more than $d/X=(1\text{.}\text{59}\text{m})/(0\text{.}\text{100}\text{m})=\text{15}\text{.}9$ because the amplitude of the oscillations is decreasing with time. At the end of the motion, this system will not return to $x=0$ for this type of damping force, because static friction will exceed the restoring force. This system is underdamped. In contrast, an overdamped system with a simple constant damping force would not cross the equilibrium position $x=0$ a single time. For example, if this system had a damping force 20 times greater, it would only move 0.0484 m toward the equilibrium position from its original 0.100-m position.

This worked example illustrates how to apply problem-solving strategies to situations that integrate the different concepts you have learned. The first step is to identify the physical principles involved in the problem. The second step is to solve for the unknowns using familiar problem-solving strategies. These are found throughout the text, and many worked examples show how to use them for single topics. In this integrated concepts example, you can see how to apply them across several topics. You will find these techniques useful in applications of physics outside a physics course, such as in your profession, in other science disciplines, and in everyday life.

Test prep for ap courses

Section summary

• Damped harmonic oscillators have non-conservative forces that dissipate their energy.
• Critical damping returns the system to equilibrium as fast as possible without overshooting.
• An underdamped system will oscillate through the equilibrium position.
• An overdamped system moves more slowly toward equilibrium than one that is critically damped.

Conceptual questions

Problems&Exercises