PHY 122 LAB : Damped Oscillations Introduction In an earlier lab we looked at a linear spring as an oscillator, which produces simple harmonic motion (SHM). A pendulum is also an oscillator, where the motion takes place along a curved path. In this lab we will measure the motion of a simple pendulum using a “sonar” detector to record the position vs time by computer. We’ll see how the period varies with pendulum length and amplitude, and analyze the motion to determine a damping constant and Q-factor. A common principle in physics when measuring very weak forces is to measure the change in frequency they cause on a nearby oscillator. (This is used in the atomic force microscope (AFM), and to test the Principle of Equivalence, and in the search for Gravity Waves). Text Reference: Wolfson 15.3,6-7.

Theory Consider a simple pendulum, with mass m and length L. Motion is constrained to the arc and the “proper” coordinate is arc length “s” or equivalently angle θ from the vertical, as shown in Fig. 1. These are related as s=Lθ (θ in radians!).

theta

L

sm

Fig. 1 Geometry of simple pendulum. For a displacement θ, there is a restoring force directed along the arc, F(θ) = -mg sin θ eq. 1

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where θ is the angle between the pendulum string and the vertical. Then we can write Newton’s 2nd law as d2θ

dt 2 +(g/L)sinθ = 0. eq. 2

where we have written acceleration along the arc as

td

dLdt

sd2

2

2

2 θ= eq. 3

It is useful to consider small angles only, so sinθ ~ θ and eq. 2 becomes

2

2

dtd θ +(g/L) θ(t) = 0. eq. 4

This has the simple sinewave solution θ(t) = θ0sin(ω0t+Φ) eq. 5 with ω0

2 = g/L eq. 6 Here θ0 is the angular amplitude and the angular frequency ω is related to the period as T = 2 π/ω0 = 2π √(L/g) eq. 7 This is fully analogous to the case for a linear spring oscillator where we had ω0

2 = k/m eq. 8

for a spring constant “k” defined by F(x) = -kx. Real oscillators have an additional force acting, namely friction, which is generally proportional (and opposite!) to velocity. This can be written as

dtdbL

dtdsbFfric

θ−=−= eq. 9

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Thus the equation of motion becomes

02

2

=+− θθθ mgdtdbL

dtdmL . eq. 10

where we have again assumed the small-angle approximation. This equation has the solution θ(t) = θ0exp(-t/τ) sin(ω’t+Φ) eq. 11 where the new frequency ω’ is slightly lower than the undamped frequency ω0 as (ω’)2 = ω0

2 – (1/τ)2 eq. 12 We define a decay time τ as τ = 2m/(bL) eq. 13 This function is a damped sinewave as shown in Fig. 2. It is usefully considered as a sinewave confined by an “envelope function” that is a simple decaying exponential f(t) = A exp(-t/τ). We can determine τ from the time for the envelope function to drop to exp(-1) = 0.368 of its maximum value. Damped oscillation is usefully characterized by a dimensionless “quality” factor given by Q=ω0τ=2πf0τ eq. 14 This can be considered as 2π times the number of cycles required for the waveform to decay by a factor of exp(-1), since we have 2π*(cycles/sec)*(sec to decay). Evidently, a large Q corresponds to slow damping (many oscillations) and vice versa.

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damped sinewave

-0.1

-0.05

0

0.05

0.1

0.15

0 5 10 15 20 25

time (sec)

thet

a(ra

d)

Fig. 2 Damped sinewave with constants: ω=2 rad/sec, tau = 9 sec, amplitude = 0.1 rad. Note the solid line representing the envelope function exp(-t/τ).

Setup: The pendulum setup is shown in Fig. 3. We will use a sonar sensor to record the position of the ball by computer. This sensor “sees” the radial distance Rs of the closest object in a cone of about 20 degrees half-angle. Check your scans to verify that it sees the object you think! We want arc length “s” but measure Rs. These are linearly related if s/R0 and s/L <<1. Thus we have θ = s/L ~ (R0-Rs)/L eq. 15

L

Rs

sonarR0

θ

s

Fig. 3 Pendulum setup with sonar position sensor.

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Procedure: 1. Measure the diameter and mass of the tennis and stryo balls 2. (T vs L) Measure the oscillation period for the tennis ball at several different string

lengths L using a stopwatch. A good scientist will instinctively realize that timing errors are reduced by counting several beats and skipping the first one (to avoid “release” errors). Use only nominal amplitudes, say about 0.1 radians.

3. (damped motion) Change to the styro ball and use a string length such that the ball is level with the sonar detector when hanging at rest. Record this length. Open the setup file “phy122/lab11”, then record a run to verify that its working. ie: see that you get a smooth oscillation and sensible values. You can over-write Run#1 etc until you get a nice one. Use nominal amplitude, say 0.1 radians. You want to record many cycles of oscillation (say 100 sec) until the motion decays to about 20% of original. Print this file for “eyeball” analysis, also copy/paste it into GA for further analysis.

Analysis: 1. Find “g” from a linearized plot of the (T, L) data and compare with the

“known” value g = 9.81 m/sec2. 2. Working in GA, make a plot of θ(t) from your Rs(t) data. (using New

Column \ Calculated). You might need to tell GA to work in radians (set preferences). Its also useful at this point to shift the time origin to start a sinewave with zero phase shift (ie: theta = 0, rising at time = 0). Eyeball your θ(t) plot to estimate the frequency, decay constant τ and quality factor Q (count the wiggles). Explain your reckoning with markings on the plot, and include some estimate of uncertainty.

3. How does the period T vary with amplitude? 4. Fit this data to a damped sinewave function (eq. 11). You may wish to

review the section on non-linear fits in “Fits.pdf” 5. Compare your eyeball vs fit values for frequency, τ and Q in a single neat

table.

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Pre-lab quiz, PHY122 Lab Your name____________________ Section day/time___________________ Find the frequency and time constant plus uncertainties for the damped sinewave data below. Do this first by eyeball, then by fits in GA. Data can be downloaded in the file “DampOsc.txt”

damped sinewave (Quiz)

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0 5 10 15 20

time (sec)

thet

a(ra

d)

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