Study of the Damped Pendulum

Study of the Damped Pendulum.
Akhil Arora, Rahul Rawat, Sampreet Kaur & P. Arun
Department of Physics & Electronics, S.G.T.B. Khalsa College,
University of Delhi, Delhi - 110 007, India∗
arXiv:physics/0608071v1 [physics.ed-ph] 7 Aug 2006
Abstract
Experiments on the oscillatory motion of a suspended bar magnet throws light on the damping
effects acting on the pendulum. The viscous drag offered by air was found the be the main contributor
for slowing the pendulum down. The nature and magnitude of the damping effects were shown to be
strongly dependent on the amplitude.
1
I.
INTRODUCTION
The simple pendulum is pedagogically a very important experiment. As a result of it’s
simplicity, it is introduced at high school level. The equation of motion (EOM) of a pendulum
undergoing simple harmonic motion (SHM) is given as
g
d2 y
= − y = −ωo2 y
2
dt
L
whose solution is easily derivable and can be taught in a class which has been introduced to
calculus. The EOM can be modified to account for damping as seen in a real pendulum and
yet the equation and it’s solution remains trivial as1
d2 y
+
dt2
b
L
!
dy
+ ωo2y = 0
dt
y(t) = e−βt (Acosω ′ t + Bsinω ′ t)
where β = (b/2mL) and ω ′ =
(1)
q
ωo2 − β 2 . However, this approach taken by textbooks over-
simplifies the complex motion of the pendulum and implies that only the pendulum’s amplitude
attenuates with time. On the contrary along with the amplitude even the oscillation’s time
period varies2 , a feature overlooked in classroom physics and carried forward for a long time
by students. The difficulty in measuring these variations also does not encourage routine
experimentation in high schools/ undergraduate laboratories. However, with the advent of
micro-computers such measurements can now be made easily. Most of the experiments reported
have measured the change in amplitude2,3,4 while examples of measuring variation in time
period is rare5 . Since, both amplitude and the time period varies with successive oscillation,
one can expect the pendulum’s velocity to vary with time at a given position. While Gregory1
used knowledge of the oscillation time period to extract information on the pendulum’s velocity,
Avinash Singh et al6 used a novel method to estimate the pendulum’s velocity. A bar magnet
was attached to a rigid semicircular aluminum frame of radius ’L’ which pivoted about the
center of the circle such that the bar magnet oscillates through a coil kept at the mean position.
As the magnet periodically passed through the coil, it generated a series of emf pulses. The
arrangement with proper circuitry determined the peak emf. Avinash et al6 approximated the
2
peak emf (ξmax ) as
ξmax ≈
dφ
dt
!
ωmax
max
where ωmax is the maximum velocity as the bar magnet passed through the mean position.
This method has it’s advantage when one proposes to study the damping effects in a pendulum. Most of the works studying the variation in oscillation amplitude4,7,8 with time have the
pendulum’s suspension connected to a variable resistance (potentiometer) which introduces
a sliding friction in the pendulum’s motion. Complex mathematics with assumption that all
damping contributors act independently is then used to filter out information of each contribution. Wang et al9 used a novel but costly method using Doppler effect to monitor the
position of the pendulum to study it’s damping. Thus, Avinash et al6 provides a interesting
yet cheap method to study the damped pendulum. While they rightly pointed out that several
parameters of the experiment such as velocity and strength of the magnet and the number of
turns in the coil can be varied, they did not explicitly discuss them theoretically or study these
factors experimentally. Hence, in this manuscript, we have furthered the study made in ref 6
and have tried to address these issues.
II.
EXPERIMENTAL SETUP
Our pendula was made by suspending a standard bar magnet by a cotton thread. The
tread was fastened to a small hook drilled into one pole of the bar magnet. The length of the
bar magnet (2l) was 7cm and the cotton thread (Ls ) used was 53cm long. A coil of 1000 turns
was kept near the pendula’s mean position at a distance ’d’ from the magnet’s lower pole (see
fig 1). The magnetic field at point ’A’ is evaluated by
1
1
µo m
−
B=
2
4π BA
CA2
(2)
where m is the dipole moment. ’AC’ and ’BA’ can be written in terms of the pendulum’s
position (angle Θ) using the cosine law. That is,
BA2 = OB 2 + OA2 − OB.OAcosΘ
AC 2 = OC 2 + OA2 − OC.OAcosΘ
3
where
O
L
Θ
B
2l
D
C
E
d
A
FIG. 1: Pendulum with the mass being replaced by a bar magnet. The detecting coil is at ’A’.
OC = Ls + 2l
OB = Ls
OA = Ls + 2l + d
hence,
BA2 = L2s + (Ls + 2l + d)2 − Ls .(Ls + 2l + d)cosΘ
AC 2 = (Ls + 2l)2 + (Ls + 2l + d)2 − (Ls + 2l).(Ls + 2l + d)cosΘ
Based on the assumption that 2l and d are relatively small compared to Ls , the higher
powers of 2l and d can be neglected. Hence, eqn(2) can be written as
"
µo M
2l
B ≈
2
4π Ls (Ls + 4l + d)(2 − cosΘ)
#
The induced emf is proportional to the rate of change in the number of magnetic lines cutting
the coil.
"
µo M
2l
dB
≈−
dt
4π L2s (Ls + 4l + d)
4
#"
#
sinΘ
dΘ
(2 − cosΘ)2 dt
Based on this, the respective induced emf can be written as
"
2l
µoMN
dB
≈
ξ = −N
2
dt
4π
Ls (Ls + 4l + d)
#"
#
sinΘ
dΘ
2
(2 − cosΘ) dt
(3)
where N is the number of turnings in the coil. Eqn(3) can be written in a compact form
ξ = ξo
"
sinΘ
(2 − cosΘ)2
#
dΘ
dt
!
(4)
where
"
µo MN
2l
ξo =
2
4π
Ls (Ls + 4l + d)
#
(5)
Thus, as the distance between the magnet and detecting coil is increased, the induced emf
decreases. Infact the induced emf is quite weak and is amplified by an op amp circuit. The
high input impedance of the IC741 opamp ensures that a true measurement of the emf is
made. To digitalise this analog signal (see fig 1, ref 6) using an Analog to Digital Convertor
ADC-0809 (see fig 2a), the amplified output is rectified and the peak value is held by charging
a capacitor. The capacitor is discharged via a large resistance so that it retains the peak value
till the next peak value arrives.
We require the ADC to start conversion once the peak value is attained by the capacitor.
This implies a synchronization between the input emf pulses and the ADC’s start of conversion
(SOC) pulses. To achieve this synchronization it is best to generate the required SOC pulse
by wave-shaping the input itself. The amplified input after rectification is fed to a comparator
which compares to +1v. This is to avoid spurious/accidental triggering due to noise. The
infinite gain results in pulses with sharp edges. The width of these pulses are approximately
To /4 (for our pendula ≈ 390ms). This would be too large for serving as a SOC and hence
is reduced to a 5µs pulse using a monostable timer made with IC55510 . The sequencing
and synchronization can be understood from the various waveforms shown in fig 2b. The
designed circuit digitalises the analog emf and on completion sends an EOC to the computer
or microprocessor kit (in case of a microprocessor this is done through a programmable I/O
IC8155 chip, details of which can be found in the book by Goankar11 ) which then reads the
eight bit data and stores it for retrivial. This project was done using an 8085 microprocessor
kit. The programme and flowchart used is detailed in the Appendix.
5
(a)
(i)
(ii)
from coil
−
+
3 (iv)
(iii)
−
+
2
555
1v +
6
22
SOC
(i)
(v)
ALE
26
data
0809
clk
10
7
EOC
(b)
from coil
+1volts
(i)
(ii)
(iii)
(iv)
(v)
t
FIG. 2: The (a) schematic diagram of the circuit used and the (b) important waveforms at the points
marked in the circuit.
The reliability of our circuit can be tested by measuring the maximum emf induced in
the coil for varying distances ’d’. Eq(4) shows that the measured maximum emf would be
directly proportional to ξo which inturn is inversely proportional to ’d’ (see eqn 5). Fig 3
6
4
3.5
ξmax
3
2.5
2
1.5
1.2
1.4
1.6
1.8
2
2.2
d (in cm)
2.4
2.6
2.8
3
FIG. 3: Variation in the maximum induced emf with increasing distance between the coil and the
magnet.
shows the variation in the experimentally determined ξmax with ’d’. While the inverse nature
is evident, the value of (Ls + 4l) as returned by curve fitting eqn(5) on our data is substantially
off mark from the actual lengths. This is expected since eqn(4) and (5) are very simplified
approximations.
III.
A.
VARIATION OF INDUCED EMF WITH INITIAL DISPLACEMENT
While undergoing undamped oscillation
The velocity of an undamped pendulum undergoing SHM is given as
dΘ
dt
where ωo =
!
q
= ωo Θ2m − Θ2
q
g/(Ls + 2l) is the frequency of oscillation and Θm is the initial displacement given
to the pendulum. Therefore, the emf induced by our pendulum undergoing undamped SHM
7
would be given as (using eq 4)

ξ ≈ ωo ξ o 
q
sinΘ Θ2m − Θ2
(2 − cosΘ)2

(6)

The variation in induced emf with time of an undamped pendulum undergoing SHM is as calculated using eqn(6) is shown in fig(4). The maximum angular displacement used to generate
the graph using eqn(6) was 5o . The emf pulse shown in fig 4 is only for half a cycle starting
from one extreme position to the opposite extreme. As the magnet approaches the coil, the
flux increases and as it crosses the mean position, the emf is negative since the magnet is
receding from the coil. Eventhough the velocity (dΘ/dt) is maximum at the mean position,
since the variation in flux (dφ/dt) is zero, the induced emf is zero as the pendulum passes the
mean position.
4
2
"
dB 0 10
dt
[2
3
sin(x)
os(x)℄2
p
2
m
x
2
0
-2
-4
-0.1
-0.05
0
(in radian)
!
0.05
0.1
FIG. 4: A measure of the induced emf with oscillating angle. The graph was generated using eqn(6)
with Θm = 5o .
The position of the pendulum when the maximum emf is generated (between 0 < Θ < Θm )
can be found as a problem of maxima and minima

h
 (2 − cosΘ)2 sinΘ d
dξ
dΘ
= ξo 
dΘ
= ξo

 (2 − cosΘ)

h
d
sinΘ dΘ
dΘ
dt
dΘ
dt
(2 −
(2 −
+ cosΘ
cosΘ)2
8
i
dΘ
dt
cosΘ)3
+ cosΘ
dΘ
dt
i
− (2 − cosΘ)sin2 Θ
− sin2 Θ

dΘ 
dt

dΘ
dt



For cases of small angle oscillations eqn(??) reduces to
"
dξ
d
= ξo Θ
dΘ
dΘ
!
dΘ
dΘ
+ (1 − Θ2 )
dt
dt

!#
(7)

q
−2Θ2
dξ
+ (1 − Θ2 ) Θ2m − Θ2 
= ωo ξ o  q
dΘ
2 Θ2m − Θ2
Θ2 = (1 − Θ2 )(Θ2m − Θ2 )
Solving the quadratic equation
Θ4 − (2 + Θ2m )Θ2 + Θ2m = 0
we have
Θm
Θpeak = ± √
2
(8)
Since eqn(7) was used to determine position of extrema, the above condition is only valid for
undamped small angle oscillations. The maxima as per this condition for magnet oscillating
through Θm = 5o occurs at ±0.0617 radians (or ±3.53o ). The maximum emf that is induced,
hence is (use eqn 6)
sin Θ√m2
Θm
ξmax = ωo ξo √ × 2
2
2 − cos Θ√m2
Since, these equations and conditions are essentially valid for small angles,
ξmax =
!
ωo ξ o
Θ2m
2
(9)
However, a physical pendulum is prone to damping and hence in the next section we investigate
as to how the maximum induced emf varies with initial displacement for a damped pendulum.
9
0.02
β=0.0
β=0.4
0.015
0.01
0.005
ξ
0
-0.005
-0.01
-0.015
-0.02
0
0.5
1
1.5
2
2.5
3
t (in sec)
FIG. 5: Variation of induced emf when oscillation is damped (β = 0.45s−1 ) is compared with the case
of no damping (i.e. β = 0.0s−1 ).
0.02
H
Y
0.015
(a)
*
(b)
0.01
0.005
?
0
-0.005
H
Y
-0.01
-0.015
-0.02
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
(in radian)
FIG. 6: The variation of induced emf of fig 3 is plotted w.r.t. angular position of the pendulum (i.e.
Θ) for the cases (a) β = 0.0s−1 and (b) β = 0.45s−1 .
B.
While undergoing damped oscillation
We have already stated in our introduction that the damped motion described by eqn(1)
exhibits how the pendulum’s oscillation amplitude decreases exponentially with time. The
10
EOM whose solution is given by eqn(1) describes a linear system. The solution can be further
trivialized without losing any generality as
Θ = Θm e−βt sin(ω ′ t)
(10)
from which the velocity can be calculated as
dΘ
= ω ′ Θm e−βt cos(ω ′t) − βΘm e−βt sin(ω ′ t)
dt
= Θm e−βt [ω ′ cos(ω ′t) − βsin(ω ′ t)]
(11)
Substituting the above expression in eqn(4) we obtain the relation showing the variation of
induced emf with time. This variation is shown in fig 5. It is also clear from the figure that
the peaks in the induced emf occurs at ωt = (2n + 1) π4 . Hence, the angles at which maxima
occur in general is written as
Θm (2n+1)π
Θpeak = ± √ e− 4tanφ
2
(12)
where tanφ = ω ′ /β. Our circuit is designed only to measure peak emfs at n=0,2,4,6....., where
only the positive solutions of eqn(12) would contribute. Using our condition on eqn(11) and
eqn(4) we have
dΘ
dt

ξpeak = (ω ′ − β)ξo 

!
Θm (2n+1)π
= (ω ′ − β) √ e− 4tanφ
2
peak
(2n+1)π
4tanφ
sin Θ√m2 e−
(2n+1)π
− 4tanφ
(2 − cos Θ√m2 e
)2



Θm − (2n+1)π
√ e 4tanφ
2
(13)
For small angle oscillations eqn(13) reduces to
ξpeak =
(ω ′ − β)ξo 2 − (2n+1)π
Θm e 2tanφ
2
(14)
The variation in emf (seen w.r.t time in fig 5) when viewed w.r.t oscillating angle Θ shows
how the peak position decreases (eq 12) as also the amplitude of the maximum induced emf
decreases (eqn 14) with each half cycle. Eqn(9) and eqn(14) shows that the maximum emf
induced for damped pendulums undergoing SHM is directly proportional to the square of
maximum angular displacement given to the pendulum. We have recorded the first maxima
11
8
7
6
ξmax
5
4
3
2
1
0
200
400
600
800
Θ2 (in degrees2)
1000
1200
1400
1600
FIG. 7: The variation of induced emf with the initial angular displacement of the pendulum. It shows
the expected parabolic dependence (i.e. Θ2m ).
reading (i.e. n=0) of the induced emf for various angles upto 40o . The linear relation between
ξmax and Θ2m is evident. Before commenting further, it must be recollected that eq(1) is valid
for small angle oscillations, i.e. for Θm < 5o , yet a good linearity is obtained till Θ = 40o .
Experimental data for Θm ≥ 45o deviate markedly from this linear trend.
Remember
eqn(12) was obtained with the assumption that the pendula’s motion is described by eqn(10).
This equation describes the motion of a pendulum oscillating in a viscous medium with small
velocity. It would be shown in the next section that the pendulum’s velocity is quite appreciable for Θm ≥ 45o and hence it’s motion is not described as in eqn(10), explaining the departure
for linearity.
IV.
RESULTS AND OBSERVATIONS
Our prelimary measurements are in good correspondence with commonly known notions
and hence we proceed to investigate further the nature of damping in our pendula. It should
be noted that the amount of damping and it’s nature are strongly pendula dependent and all
results reported here are specific to our experiment and can not be taken as general. We have
12
1
(a)
(b)
(c)
(d)
0.9
ξ (Normalized)
0.8
0.7
0.6
0.5
0.4
0
20
40
60
80
n
100
120
140
160
FIG. 8: The variation in peak induced emf measured with each oscillation is shown for initial displacements (Θm ) (a) 5o , (b) 30o , (c) 55o and (d) 65o .
recorded the maxima in induced emfs for 80 oscillations for various initial displacements. Since
for each oscillations, we get two positive maxima in induced emf, figure 8 shows the variation
in maxima reading of induced emf for 160 peaks.
A general expression ae−bn was fitted to the data of ξmax w.r.t n using a standard and freely
available curve fitting software called ”Curxpt v3.0”. Good fits were obtained for oscillations
set by initial displacements upto 40o . The exponential fall in emf is indicative of the rate of
loss of energy from the oscillating system. It indicates the loss to follow the relation
dE
∝ −E
dt
or
dE
= −bE
dt
This indicates that the velocity is low and hence the damping/resistive force acting on the
pendulum is proportional to the velocity12 . Figure 9 shows the variation of the decay constant
’b’ (of ae−bn ) with respect to the maximum displacement (Θm ) given to the pendulum. The
graph indicates that as Θm increases, the velocity with which the pendulum moves increases
with which the damping constant increases.
13
0.005
0.0045
Damping Constant
0.004
0.0035
0.003
0.0025
0.002
0.0015
0.001
0
10
20
30
40
50
60
70
θm
FIG. 9: The variation decay constant (b of general equation ae−bn ) of the exponentially decaying
√
region. The continuous line is the parabolic fit for the data points (0.00056 Θm ).
0.25
Decay Constant
0.2
0.15
0.1
0.05
0
45
50
55
60
65
70
θm
FIG. 10: The variation decay in ’b’, a measure of decay in the early oscillations when pendulum was
set in motion with displacements > 45o .
For displacement angles beyond 45o (Θm ≥ 45o ), eventhough visual examination of the
curves in fig 8 suggests an exponential damping, the data points do not fit to an exponential
fall relation. A detailed examination suggests a more complex process is taking place with
initial damping being sharper. Infact the initial 25-30 data points fit to a/nb . The data
14
beyond this fit to the exponential fall equation. The a/nb fit corresponds to the damping force
being proportional to higher powers of velocity (vγ , where γ > 1) and in turn the rate of energy
loss also being proportional to higher terms of energy. That is, the rate of energy loss for our
pendulum set into oscillations with a displacement angle > 45o is given as
1+b
dE
= −αE b
dt
Figure 10 plots ’b’ versus Θm . The power term (b) is being treated as a measure of damping
and is consistent with the results of fig 9, i.e. as initial displacement increases the damping
becomes large with a proportionality to the pendulum’s velocity.
The resistive force being proportional to higher powers of velcoity has been reported earlier
also3,4,13,14,15,16 . A system is reported to have a constant friction (γ = 0) or a linear dependence
of velocity (γ = 1) or a quadratic depenence of velocity (γ = 2). Corresponding to which
the pendulum’s amplitude decays linearly, exponentially and inverse power decay respectively
with time. It hence maybe concluded that for our pendulum sent into motion by initial
displacements Θ ≥ 45o , the damping force is proportional to vγ where γ > 1.
V.
CONCLUSION
A simple experiment of setting a suspended bar magnet into oscillations, is a rich source of
information. Not only does it give exposure to Faraday’s induction law and a basic understanding of induced emf’s dependence on angle of oscillation, it enables us to study the damping
effects on the pendulum. This method is better than previously used methods since the measuring technique does not introduce additional contributions to damping. When the oscillation
imparted to the pendulum is very large, the damping effect is also strong with the damping
force being proportional to vγ , where γ > 1 and ’v’ is the pendulum’s velocity. This brings
down the oscillation amplitude of the pendulum and it’s velocity. As the velocity becomes
low, the resistive force acting on the pendulum changes it’s nature and becomes proportional
to ’v’. Considering the rich information obtained from the experiment and the simplicity of
the experiment, it allows the method to be easily implemented as a routine experiment in
undergraduate laboratories.
15
Acknowledgements
The authors would like to express their gratitude to the lab technicians of the Department
of Physics and Electronics, SGTB Khalsa College, for the help rendered in carrying out the
experiment.
∗
Electronic address: arunp92@physics.du.ac.in
1
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2
M. F. Mclnerney, ”Computer-aided experiments with the damped harmonic oscillator”, Am. J.
Phys., 53, 991-996 (1985).
3
A. R. Ricchiuto and A. Tozzi, ”Motion of a harmonic oscillator with sliding and viscous friction”,
Am. J. Phys., 50, 176-179 (1982).
4
Patrick T. Squire, ”Pendulum Damping”, Am. J. Phys., 54, 984-991 (1986).
5
Neha Agarwal, Nitin Verma and P. Arun, ”Simple Pendulum revisited”, European. J. Phys., 26,
517-523 (2005).
6
Avinash Singh, Y. N. Mohapatra and Satyendra Kumar, ”Electromagnetic induction and damping:
Quantitative experiments using a PC interface”, Am. J. Phys., 70, 424-427 (2002).
7
L.F.C. Zonetti, A.S.S. Camargo, J. Sartori, D.V de Sousa and L.A.O. Nunes, ”A demonstration
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8
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9
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10
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11
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16
12
Avinash Singh, arXiv:physics/0206086.
13
B. J. Miller, ”More Realistic Treatment of the Simple Pendulum without Difficult Mathematics”,
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14
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15
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17
Appendix
Table 1. Program used to collect data.
Address Mnemonics Hex Code Address Mnemonics Hex Code
C400
LXI SP
31
C40A
01H
01
C401
00H
00
C40B
JZ
CA
C402
C3H
C3
C40C
07H
07
C403
MVI A
3E
C40D
C4H
C4
C404
00H
00
C40E
IN
DB
C405
OUT
D3
C40F
09H
09
C406
08H
08
C410 PUSH PSW
F5
C407
IN
DB
C411
JMP
C3
C408
OBH
OB
C412
07H
07
C409
ANI
E6
C413
C4H
C4
18
Start
Initialise Stack Pointer
Set Control word making
Port A and Port C inputports
Read port C, (contains EOC)
AND content with 01H
Is EOC high
No
Yes
Read Data at Port A
Push PSW to save data
Unconditional Jump
FIG. 11: Flowchart.
19