03 Damped Harmonic Motion

Aim¶

To show the effect of damping on amplitude and frequency of an oscillation.

Subjects¶

• 3A50 (Damped Oscillators)

Diagram¶

Equipment¶

• Track, $2.2\mathrm{~m}$.

• Cart.

• Motion sensor, mounted on cart.

• Reflecting screen.

• Two springs.

• Screen, ($50\times 50\mathrm{~cm^2}$; $m=0.5\mathrm{~kg}$) with socket screw.

• Socket head to mount screen on cart.

• Mass $m=0.5\mathrm{~kg}$.

• Pair of scales with large display.

• Photo-gate and laser.

• Data-acquisition system and software.

• Beamer to project the monitor-screen.

Presentation¶

Mount the cart with motion sensor and the mass of $.5 \mathrm{~kg}$ between the two springs that are attached to the end-stops of the track. Position the reflecting screen, needed for the motion sensor, in front of the cart.

Give the cart a deflection, start the data-acquisition system and let the cart go. Data are collected during 20sec.

Remove the mass of $.5 \mathrm{~kg}$. Show by means of a pair of scales that the $50 \times 50 \mathrm{~cm}^{2}$ screen has also a mass of $.5 \mathrm{~kg}$. Mount the screen on the cart. Give the cart the same deflection as before and measure again during 20sec.

The two graphs of position can be studied and discussed now (see Figure 3). Clearly can be observed that the screen on the cart introduces more damping to the oscillating system. Also can be observed that the frequency of the damped oscillator is slightly lower than the frequency of the undamped oscillator.

Explanation¶

Damping happens due to resistance forces dissipating energy. Such forces can be described assuming that the magnitude of the resistance force is related to the speed of the body as $F=-b v^{n}$ ($n$ ranges from 1 to 2).

For many situations $n$ is given the extreme value of $n=1$, making the resistance force equal to $F=-b v$. Then for such a damped oscillator the position of mass $m$ can be expressed by $x=e^{-\alpha t}\left(A \cos (\omega t)+B \sin (\omega t)\right)$ with $\alpha=\frac{b}{2 m}$.

The angular frequency of a damped system equals $\omega=\sqrt{\omega_{0}^{2}-\frac{b^{2}}{4 m^{2}}}, \omega_{o}$ being the frequency in the absence of damping. So increasing $b$ means decreasing $\omega$, hence the period of damped oscillation is larger than the period of the undamped oscillation.

Sources¶

• Alonso, M/Finn, E. J., Fundamentele Natuurkunde, part 1, Mechanica, pag. 297-299

• Mansfield, M and O’Sullivan, C., Understanding physics, pag. 99-101

• Roest, R., Inleiding Mechanica, pag. 266-274

• Young, H.D. and Freeman, R.A., University Physics, pag. 411-412