MC-15b  Simple Harmonic Motion and Resonance (Air Track)

OBJECTIVES:

1. To study the period of Simple Harmonic Motion (SHM) as a function of oscillation amplitude.
2. To study the period of SHM as a function of oscillating mass.
   Expected result: Period( $\equiv T = 2\pi/\omega = 1/f$)
   is proportional to $\sqrt{\mbox{Mass/Spring Constant($k$)}}$
3. To demonstrate Hooke's Law, $F=-kx$
4. To observe the relationships between the potential and kinetic energy

THEORY:

The restoring force ($F$) on an object attached to a ``simple'' one-dimensional spring is proportional to the displacement from equilibrium and has the form, $F= -k(x-x_0)$, where $k$ is the spring constant (or stiffness in N/m), $x_0$ is the equilibrium position (i.e., no net force) and $x$ is the position of the object. This is Hooke's Law. Remember that the simple harmonic oscillator is a good approximation to physical systems in the real world, so we want to understand it well. That's the purpose of this lab!

The expression $F= ma = -k(x-x_0)$ is a 2nd order differential equation with

$$a =\frac{d^2 x'}{dt^2}= -\frac{k}{m} (x')$$

where $x'\equiv x-x_0$. The most general solution for this expression is often given as

$$x'(t)= A \cos(\omega_0 t) + B \sin(\omega_0 t)$$

where $\omega_0\equiv \sqrt{k/m}$. $\omega_0$ is defined as the natural frequency for the undamped harmonic oscillator. $A$ and $B$ are arbitrary initial displacement parameters. Alternatively the solution is often specified as

$$x'(t)= C \sin(\omega_0 t + \phi) \mbox{     or     } x'(t)= C \cos(\omega_0 t + \phi)$$

where $\phi$ is the starting phase and $C$ is the displacement. It is also possible to write down these solutions using complex numbers as in

$$x'(t)= A' e^{i\omega_0 t} + B' e^{-i\omega_0 t}  .$$

The relative merit of these equivalent expressions will become clearer in EXPTS. V and VI.

FUNDAMENTAL CONCEPTS:

1. The solution to the undamped (i.e., no frictional or drag forces) harmonic oscillator is time dependent and periodic. When the $\omega_0 t$ term varies by $2\pi$ (or one period $T = 1/f = 2\pi/\omega_0$) both the position, $x'(t+T)=x'(t)$, and velocity, $v'(t+T)=v'(t)$ return to their previous values.

2. Total energy (TE) is conserved in SHM motion. As time evolves kinetic energy (KE) is transferred to (and from) potential energy (PE). Thus at any time $t$:
   
   $$TE = \mbox{constant} = KE(t) + PE(t) = \frac{1}{2}m [v'(t)]^2 + \frac{1}{2}k [x'(t)]^2     .$$
   
   

APPARATUS:

Basic equipment: Air track, assorted slotted masses, air supply, hose, adjustable stop, glider, springs, timer, photogate & support stand, knife edge assembly.

Computer equipment: Personal computer set to the MC15A lab manual web-page; PASCO interface module; photogate sensor and extension jack, PASCO sonic position sensor, speaker with driver stem, power amplifier module.

PRECAUTIONS:

1. See MC-14b
2. When setting up the springs, use the adjustable end stop on the air track so the springs are stretched properly: don't stretch them beyond their elastic limit and don't let them sag and drag on the track.
3. Keep the amplitudes small enough that a slack spring doesn't touch the track nor a stretched spring exceed its elastic limit.

SUGGESTIONS: To measure the period of the oscillating glider:

1. Locate the photogate so the glider just cuts off beam when glider is in the equilibrium position ($x-x_0 = 0$). The photogate phone jack should be in the first PASCO interface position. Choose two springs having a similar length and refer to the above precaution. Turn on the air supply adjust the blower speed to minimize frictional forces.

2. To initiate the PASCO interface software click the computer mouse when centered on the telescope icon in the ``toolkit'' area below. There will be a just a single table for recording the measured period.

3. Start the glider by displacing it from equilibrium and then releasing it. Then start the data acquisition by clicking the REC button button. Let the glider oscillate for about 10 periods. Calculated the mean and standard deviation by simply clicking on the statistics icon (i.e. ) on the data table.

Figure 1: Sketch of the air track configuration for SHM.

UNDAMPED SIMPLE HARMONIC MOTION:

1.

Using sketches in your lab book, both at equilibrium and after a displacement from equilibrium, show that the effective force constant for 2 identical springs of force constant $k$ on either side of an oscillating mass is $2k$.

2.

EXPERIMENT I: Show experimentally that the period is independent of the amplitude. Try amplitudes of approximately 10 cm, 20 cm, and 30 cm. (Friction may be a problem at very small amplitudes).

3.

EXPERIMENT II: By adding mass to the glider, study the period versus total oscillating mass. The latter must include a correction for the oscillating springs whose effective oscillating mass (see the note below) is approximately $m_s$/3 where m$_s$ is the mass of the two springs. Explain, in words, how this correction may be qualitatively justified (Why not $m_s$ or $m_s/2$?) By inspection of your $T$ vs $[M + (m_s/3)]$ curve, what function of $T$ might yield a straight line when plotted versus $[M + (m_s/3)]$? Prepare this plot. Calculate the effective force constant, $k'$, (equal to $k_1+k_2$, the two springs will actually differ slightly) of the system from the slope of this straight line graph. (Remember $T = 2\pi \sqrt{M_{\mbox{\tiny eff}}/k'}$ where $M_{\mbox{\tiny eff}} \; \simeq$ M + m$_s$/3.) Estimate the uncertainty in $k'$ by solving for $k'$ at each glider mass and calculating the standard deviation of the mean.

NOTE: Theoretically $m_{\mbox{\tiny eff}}=m/3$ only for $M/m_s=\infty$ For $M = 0$, the effective mass is $4m/\pi^2= 0.405$ m. However for $M/m_s$ = 5, the effective mass is already $\simeq 0.336 \; m_s$. See Fig. 1 of J.G. Fox and J. Makanty, American Journal of Physics, 38, 98 (1970).

4.

EXPERIMENT III : Also measure $k$ directly by hanging the two springs vertically on a ``knife edge'' assembly in the laboratory. Record the stretch produced by a series of weights, but do not exceed the elastic limit of either spring! Graph $F$ vs $y$ for each spring and obtain a best-fit straight line. The slope should be the spring constant. Compare this value of $k'$ with that obtained in part #3.

5.

QUESTION: Assuming the spring constant doubles, how would T vary?

FURTHER INVESTIGATIONS OF ``UNDAMPED'' SIMPLE HARMONIC MOTION:

Up to this point you have characterized the SHM of a spring-mass assembly in terms of only a single parameter $T$ (the period). One of the possible time-dependent expressions for describing the motion was

$$x'(t)= A \cos(\omega_0 t +\phi_0)   \mbox{and}   v'(t)= -A\omega_0 \sin(\omega_0 t +\phi_0)$$

or

$$v'(t)=  A\omega_0 \cos(\omega_0 t +\phi_0-\frac{\pi}{2}) \hspace*{2in}$$

where $A$ is the amplitude (displacement from equilibrium), $\omega_0$ is the natural angular frequency and $\phi_0$ is the starting phase. Notice that the velocity has a phase shift of $\frac{\pi}{2}$ relative to the displacement.

Equally characteristic of SHM is the process of energy transference: kinetic energy of motion is transferred into potential energy (stored in the spring) and back again. Friction is an ever present energy loss process so that the total energy always diminishes with time.

To capture this rather rapid cyclic process we will again use the PASCO interface while replacing the photogate sensor with the sonic position sensor.

EXPERIMENT IV: Measuring the $x$ vs $t$ behavior:

1. Remove the 10 cm timing plate and plug the aluminum vane into the central banana plug position with the vane perpendicular to the long axis of the glider. The mass of this arrangement will now have changed slightly; predict the new natural frequency.

   Figure 2: Sketch of the air track with the position sensor.

   

2. Place the position sensor approximately 60 cm from the vane in the direction of oscillatory motion. Make sure the yellow phone jack is in the third slot and the black phone jack is in the fourth slot. Alignment is very important so that the sensor senses only the vane and not the cart. A slight upward tilt may help (or raising the vane up slightly as well).

3. CLICK on the telescope icon below to initiate the PASCO $^{\mbox{\copyright}}$ interface software.

4. Displace the cart approximately 20 cm from equilibrium to initiate the oscillatory motion. CLICK on the the REC button to start your data acquisition. The graph will simultaneously display both absolute position and velocity versus time.

5. Practice a few times to make sure you can obtain smoothly varying sinusoidal curves. Then run the data acquisition for just over ten cycles and use the cross-hair feature to read out the time increment for ten full cycles. How does your prediction check out? Determine the initial phase (i.e., at $t=0$). Record the equilibrium position ($x_0$) as well.

6. Use the magnification option of the PASCO software to better view a single full cycle by clicking on the magnifying glass icon (in the graph window) and then select two points in the graph using a CLICK and DRAG motion of the mouse.

7. Print out (click in graph region and then type ALT, CTRL-P) or, alternatively, sketch the position and velocity curves in your lab book identifying key features in the time dependent curves. In particular identify the characteristic(s) which demonstrate the $\pi/2$ phase difference between the velocity and displacement curves.

8. Using the PASCO cross-hair option to read out the relevant time, position and velocity, make a table as below and fill in the missing entries (identify units).

   $$\omega_0 t+\phi_0 x'(t) v'(t) KE PE TE$$ time 0            

   $$\pi/4$$

   $$\pi/2$$

   $$3\pi/4$$

   $$\pi$$

9. Is the total energy a constant of the motion?

10. How much displacement amplitude and energy are lost after five full cycles? What is the approximate friction coefficient?

OPTIONAL INVESTIGATIONS OF ``UNDER-DAMPED'' SIMPLE HARMONIC MOTION:

In the real world friction is an ever present process. In the case of SHM friction friction can have a profound effect. Damping of unwanted vibrations is important in a myriad of situations. (Imagine what driving a car would be like if there were no shock absorbers!)

Introducing friction can be done by simply adding one more term in the force expression, $F_{\mbox{\small drag}} \equiv -R v$, a drag force which is proportional to the velocity where $R$ is the drag coefficient [units of kg m/s or N/(m/s)] . This is appropriate for motion thru a viscous fluid but it is really only a rough approximation for the frictional forces in the air track. The modified force expression now becomes

$$F= ma = -k x' - R v'   \mbox{or}   0=\frac{d^2x'}{dt^2}+\frac{R}{m} \frac{dx'}{dt}+\omega^2 x'$$

Since energy is continually lost solutions of this expression will be time dependent but NOT periodic. Adding this ``simple'' term dramatically complicates the process of finding appropriate solutions. The most general form of the solution is

$$x' = e^{-(R/2m)t}  \left[ A e^{+\sqrt{(\frac{R^2}{4m^2}-\omeg... ...}  t} + B e^{-\sqrt{(\frac{R^2}{4m^2}-\omega_0^2)}  t}\right]$$

which is quite formidable. Since the air track drag is low (i.e., R is relatively small) the solution is said to be underdamped and oscillatory when $\omega _0^2 > R^2/4m^2$. The solution in this case becomes:

$$x' = C e^{-(R/2m)t}  \cos(\omega_1t+D)   \mbox{where}   \omega_1\equiv \omega_0^2-\frac{R^2}{4m^2}$$

where $\omega_1$ is the natural frequency of the underdamped system and C and D are the initial displacement and phase. Because the frequency is lowered, the period lengthens. This is consistent with one's intuition; drag works against oscillatory motion.

Figure 3: Sketch of underdamped harmonic motion with $\omega _0^2 > R^2/4m^2$.

EXPERIMENT V:

1. Reduce the air flow to the air track so that the amplitude of the glider motion diminishes by 50% (or so) in a few minutes. Typically the lowest blower setting will work well enough. (Make sure the other group is also ready.)

2. CLICK on the telescope icon below to initiate the next PASCO $^{\mbox{\copyright}}$ interface application. In addition to the graph there will be a ``two'' column table displaying the time and position then the time and velocity.

3. Displace the cart approximately 20 cm from equilibrium to initiate the oscillatory motion. CLICK on the the REC button to start your data acquisition and record enough cycles to see the amplitude diminish by two-thirds.

4. Repeat the experiment at full blower speed for approximately the same period of time. Can you distinguish the difference between ``$\omega_0$'' and $\omega_1$ ($T_0$ and $T_1$)?
   HINT: The graph display for the position is configured to overlay the data sets. Use the magnifying glass icon feature to examine the relative phase difference at early time and late time.

5. If so, from the formula $\omega_1^2=\omega_0^2-R^2/4m^2$ determine $R$.

6. For the reduced air flow data, select six representative times using the table (identifying where the velocity changes sign) and make a table of $t$ vs maximum displacement. Use these points in the graphical analysis package and fit these points to the expression $C \exp{-(R/2m)t}$ or, in terms of the explicit analysis formula, $y= C *\exp[-B*(x-x_0)]$.

7. Does this value compare favorably with the results of item 4? How good or poor is the assumption that the drag force is proportional to the velocity?

8. Sketch out an approximate curve of $x$ vs $t$ if $R$ were significantly larger. Which $R$ would be more appropriate for absorbing and dissipating a physical ``shock'' (and why)?

OPTIONAL INVESTIGATIONS OF RESONANCE:

One of the most important situations of the harmonic oscillator is that of FORCED, damped harmonic motion. In one-dimension the applied force is typically sinusoid and when $\omega$ approaches the natural frequency of the system (nominally $\omega_1$ and, if $R^2/4m^2 \ll 1$, also $\omega_0$) the energy of the driver is additively coupled to the moving mass (e.g., a glider) and resonance occurs. Resonance is a very important aspect of the world around us and many mechanical and electronic devices employ resonant behavior as a fundamental aspect of their operation (e.g., musical instruments, radios, televisions).

Along with the frictional drag ($R v'$ where $R$ is the drag coefficient) one more force term must be added, that of the mechanical driver, with $F_{\mbox{\small driver}}= F_d \cos(\omega t)$. The new force expression is conventionally written as:

$$F= ma = - k x' - R v' + F_d \cos \omega t     \mbox{or}     ... ... =\frac{d^2x'}{dt^2}+\frac{R}{m} \frac{dx'}{dt}+\omega^2 x'   .$$

Since the system energy is lost through friction and may be gained through the driver action, solutions of this expression will be time dependent but with both transient and steady-state attributes. In many instances resonant systems respond so quickly that one only views the steady-state behavior. In this lab you will be able to observe BOTH the transient and steady-state processes.

As you may expect the most complete solution of this new differential equation has a rather complicated form and so is not reproduced here. Since we are interested only in resonance we can simplify the expression by assuming solutions that apply to the underdamped case (those with oscillatory behavior). Thus the analytic solution reduces to:

$$x' = Ce^{-(R/2m)t}  \cos(\omega_1t+D) + \frac{F_d}{[m^2(\omega_0^2-\omega^2)^2+\omega^2 R^2]^{1/2}} \cos(\omega t - \phi)$$

where the first term is the transient behavior, identical to that of the simple damped harmonic oscillator (described in the last section) and the second term is the steady-state solution. At large times $t$ the first terms dies out exponentially so that $x'$ is approximated by only

$$x'((R/2m)t \gg 1) = \frac{F_d}{[m^2(\omega_0^2-\omega^2)^2+\omega^2 R^2]^{1/2}} \cos(\omega t - \phi)$$

where $\omega_0$ is the natural frequency of the undamped harmonic oscillator, $\omega$ is the mechanical driver frequency, $R$ is the drag coefficient and $\phi$ is a measure of the phase difference between the driver motion and the glider motion.

In this lab we will only investigate the nature of the glider displacement with driver frequency ($\omega$) in the vicinity of the resonant frequency. Thus the only relationship of interest becomes $x' \propto [m^2(\omega_0^2-\omega^2)^2+\omega^2]^{-1/2} \equiv Z^{-1/2}$. $Z$ is a minimum when the driver frequency is set to $\omega_0^2- R^2/(2m^2)$ or $\omega_1^2 - R^2/(4m^2)$ which is defined to be $\omega_2$. Since $R^2/2m^2$ is small (the air-track is a low friction experiment) $\omega_0$ is nearly the same and, in addition, $x'$ will be sharply peaked about $\omega_0$.

Figure 4: Sketch of the relative maximum displacement squared vs driver frequency.

SUGGESTED PROCEDURE:

1. EXPERIMENT VI: Make sure the 10 cm timing plate is replaced by the aluminum vane (in the center banana plug position) with the vane perpendicular to the long axis of the glider. Predict the natural frequency for this new arrangement.

   Figure 5: Sketch of the air track with the position sensor and speaker.

   

2. Place the position sensor approximately 60 cm from the vane in the direction of oscillatory motion. Make sure the yellow phone jack is in the third slot and the black phone jack is in the fourth slot. Alignment is very important so that the sensor senses only the vane and not the cart. A slight upward tilt may help (or raising the vane up slightly as well).

3. Detach the fixed spring end stop and place the speaker as shown in the figure above with the spring looped through the small slot in the speaker driver stem using the same considerations for the spring extension as in the previous experiments.

4. Make sure the speaker power leads are plugged into the amplifier module and that its power is turned on. Also verify that the DIN-9 pin connector (from the amplifier module) is plugged into the A position in the PASCO interface module.

5. CLICK on the telescope icon below to initiate the PASCO $^{\mbox{\copyright}}$ interface software.

6. Make sure the speaker driver window has been switched to the Off position. With maximum blower airflow, displace the cart approximately 20 cm from equilibrium to initiate the oscillatory motion. CLICK on the the REC button to start your data acquisition. The graph will simultaneously display glider position and velocity versus time and the amplifier current (which should be zero). Practice a few times to make sure you can obtain smoothly varying sinusoidal curves.

7. Reduce the air flow to the air track so that the amplitude of the glider motion diminishes by 50% (or so) in a few minutes. Typically the lowest blower setting will work well enough. (Make sure the other group is also ready.)

8. Displace the cart approximately 20 cm from equilibrium to initiate the oscillatory motion. CLICK on the the REC button to start your data acquisition and record enough cycles to see the amplitude diminish by 90%. Compare your data to Fig. 3 and verify that your glider has a similar transient behavior. Determine the natural frequency of the system and compare to your prediction. (This frequency is actually $\omega_1$ but it is, for this low friction set-up, nearly the same as either $\omega_2$ or $\omega_0$.) To observe the steady-state properties of resonance you will have to wait for times longer than those required for this step.

9. Moving to the PASCO Signal Generator window (shown in Fig. 6), set the amplifier frequency (which is in Hertz or cycles per second) to the closest to resonance (0.01 Hz steps) and engage the speaker but CLICKing the Auto icon. The driver output should use the sinusoid AC waveform and, if the overload light on the Power Amplifier flashes on, reduce the voltage setting slightly. Start the data acquisition (and speaker motion) by CLICKing on the REC button and record data until you achieve steady-state behavior.
   

   NOTE: The nominal step sizes for adjusting the amplifier frequecy and voltage are very large. To reduce the step size depress either the CTRL key (1 Hz), the ALT key (0.1 Hz), or both (0.01 Hz) while simultaneously CLICKing the Up or Down arrows.

   Figure 6: Sketch of the PASCO software Signal Generator window.

   

10. Set the amplifier frequency 0.01 Hz steps above and below resonance and record data until you achieve steady-state behavior. Repeat for 0.10 Hz and (if time permits) 0.40 Hz steps. Plot out a few representative sets of data.

11. Determine the maximum steady-state displacement of the glider for each of the measured frequencies and plot the relative amplitude squared [( $x'(\omega)/x'(\omega_0))^2$] vs frequency offset ( $\omega-\omega_0$). Estimate the full width at half maximum for this curve. This value should be equal to $R/m$.

12. Discuss the nature of this resonance curve. If you adjust the $R/m$ ratio to further sharpen the resonance curve can you identify a compensating complication if you are interested in achieving steady-state behavior? If the speaker were attached to an amplifier playing audible music (nominally 20-20,000 Hz), what do you think this the mass will do?