Damping in a variable mass on a spring pendulum Rafael M. Digilov,a兲 M. Reiner, and Z. Weizman Department of Education in Technology and Science, Technion-Israel Institute of Technology, Haifa 32000, Israel
共Received 8 December 2004; accepted 27 May 2005兲 A model of damped oscillations of a variable mass on a spring pendulum, with the mass decreasing at a constant rate is presented. The model includes damping terms that are linear and quadratic in the velocity. The effect of the mass loss rate on the magnitude of the damping parameters is discussed. The model is compared to experimental data obtained with the aid of computer-based laboratory equipment. © 2005 American Association of Physics Teachers. 关DOI: 10.1119/1.1979498兴 I. INTRODUCTION Inexpensive computer-based equipment was recently used to measure the damped oscillations of a spring pendulum whose mass decreases at a constant rate,1 共1兲
m共t兲 = m0 − rt,
where m0 is the initial mass of the pendulum and r is the mass loss rate. The damping due to energy loss is caused by the mass loss and air drag. In Ref. 1 the air drag was assumed to be proportional to the velocity of the motion, 共2兲
Fd = − bv ,
where b is the damping constant, and an approximate expression was obtained for the decay of the oscillation amplitude as a function of time:1
冉
A共t兲 = A0 1 −
r t m0
冊
b/2r+1/4
.
共3兲
The ratio, A共t兲 / A0, is independent of the initial amplitude A0. Three characteristic decays of the oscillation amplitude were identified, depending on which damping mechanism dominates.1 When air resistance plays the dominant role 共b Ⰷ r兲, the envelope is an exponential; when b ⬃ r, the envelope is linear; and when the energy loss due to the mass loss dominates 共b Ⰶ r兲, the envelope is convex. For the linear air drag model to be valid, the amplitude of the oscillations should be much smaller than the characteristic dimension of the body and the Reynolds number Re should satisfy Re艋 3000.2 In most cases this approximation does not account for the observed behavior of real pendulum, and nonlinear damping terms are generally necessary.3,4 In this paper we describe the damped oscillations of a variable mass spring pendulum for Re⬎ 104 by including an additional damping term due to air drag that is quadratic in the velocity. The approximate solution for the decay of oscillation amplitude is compared to data obtained with the aid of computer-based equipment.
II. THE MODEL Consider a container filled with sand and suspended from a coil spring with force constant k. The container oscillates along the vertical axis z and its mass m共t兲 decreases as in Eq. 共1兲. The law of momentum conservation for a variable mass system is5 901
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dm d , 共mv兲 = F + u dt dt
共4兲
where v = dz / dt, z increases downward, and u共dm / dt兲 is the rate at which the momentum leaves the system. The net external force, F, on the container is due to its weight Fg = m共t兲g, where g is the acceleration of gravity and the restoring force, which is proportional to the displacement of the spring from its initial position z = 0, Fk = −k共z + z0兲, where z0共t兲 = m共t兲g / k is the equilibrium position of the spring. The air drag, Fd, is assumed to be a sum of terms that are linear and quadratic in the velocity Fd = − b共dz/dt兲 − c兩dz/dt兩共dz/dt兲,
共5兲
where c is the quadratic damping parameter. If we assume that the velocity of the sand relative to the container is zero at the moment it leaves the container, then u ⬇ v = dz / dt and Eq. 共4兲 becomes m共t兲
冏冏
dz dz dz d 2z + b + kz = 0. +c dt dt2 dt dt
共6兲
Equation 共6兲 is nonlinear and cannot be solved analytically. Our approximate solution is based on the assumption that the damping is small and its effect on the change in the amplitude A and the angular frequency over a single oscillation is small compared to the amplitude A0 and the frequency 0 ⬇ 冑k / m0 without damping. For this assumption to be a valid, we must have b Ⰶ 冑km, c Ⰶ m / A, and r Ⰶ 冑km. The first approximation to the solution of Eq. 共6兲 can be written as z共t兲 = A共t兲cos共t + 兲,
共7兲
where A共t兲 slowly decreases, and 共t兲 ⬇ 冑k / m共t兲 slowly increases due to the mass loss; is the initial phase. The functional form of A共t兲 can be found from the energy loss equation for a variable mass system, which in our case 共u ⬇ v兲 can be derived4 by taking of the scalar product of v with both sides of Eq. 共4兲. The result is dE v2 = vFd − r . 2 dt
共8兲
The energy loss over a single cycle is obtained by integrating Eq. 共8兲 over the period T = 2 / from t to t + T. With Eq. 共5兲, we have © 2005 American Association of Physics Teachers
901
1 ⌬E共t兲 =− T T
冕 冋冉 冊 t+T
b+
t
册
r 2 v + cv3 dt. 2
共9兲
Because A共t兲 and 共t兲 vary slowly over a single oscillation, we have from Eq. 共7兲 v = dz/dt ⬇ A共t兲共t兲sin共t + 兲 = X共t兲sin共t + 兲,
共10兲
where X共t兲 = 共t兲A共t兲. We substitute Eq. 共10兲 in Eq. 共9兲 and obtain
冉 冊
r ⌬Et = − b + 具v2典 − c具v3典, T 2
共11兲 Fig. 1. Schematic of the experimental setup for monitoring the damped oscillations of a variable mass on a spring pendulum.
where 具v 典 denotes the average over a period: n
具 v n典 =
关X共t兲兴n T
= 关X共t兲兴n
冕 冕
t+T
关sin共t + 兲兴ndt
A共t兲 =
t
2
/2
0
2 共sin 兲nd = 关X共t兲兴n In .
共12兲
We changed variables, = t, and integrated over a quarter instead of a complete period. We define In = 兰0/2共sin 兲n d such that In = / 4, for n = 2, and In = 2 / 3, for n = 3. Because A共t兲 decreases slowly, the total energy E共t兲 over a single period is conserved and can be written as E共t兲 =
m共t兲 m共t兲 关A共t兲共t兲兴2 = 关X共t兲兴2 , 2 2
共13兲
or dE共t兲 dm dX 1 = m共t兲X共t兲 + 关X共t兲兴2 . dt dt 2 dt
共14兲
共15兲
and equate the right-hand sides of Eqs. 共11兲 and 共14兲 and obtain m共t兲
冉 冊
4c 1 r dX共t兲 关X共t兲兴2 . = − b − X共t兲 − 2 dt 2 3
共16兲
We then integrate Eq. 共16兲 with initial condition X共0兲 = A00 to give X共t兲 =
A00共1 − rt/m0兲␣ , 1 + 关1 − 共1 − rt/m0兲␣兴
共17兲
where ␣ and  are
␣=
b 1 − , 2r 4
共18兲
=
16A00c . 3共2b − r兲
共19兲
Because X共t兲 = 共t兲A共t兲 = A共t兲0共1 − rt / m0兲−1/2 where 0 = 冑k / m0, Eq. 共17兲 becomes 902
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共20兲
which expresses the decay of the amplitude over long-time scales compared to the period of oscillation. Unlike the linear damping case,1 the ratio A共t兲 / A0 in Eq. 共20兲 depends on the initial amplitude A0 and the angular frequency 0. In the limit r → 0, Eq. 共20兲 yields the damping equation for a fixed mass pendulum A共t兲 =
A0 exp共− ␣0t兲 1 + 0共1 − exp共− ␣0t兲兲
共r → 0兲,
共21兲
with ␣0 = b0 / 2m0, and 0 = 8A00c0 / 3b 共the subscript zero means that damping parameters are defined for r = 0兲. In the limit r → ⬁, Eq. 共20兲 yields A共t兲 = A0共1 − rt/m0兲1/4 ,
Because the energy loss over a period is small, we can use the approximation ⌬E dE ⬇ , T dt
A0共1 − rt/m0兲␣+1/2 , 1 + 关1 − 共1 − rt/m0兲␣兴
共22兲
that is, for a large mass discharge rate, the damping is governed by the mass loss. In this case A共t兲 / A0 does not depend on A0 and 0 and has a convex envelope. We will fit the experimental data to Eq. 共20兲 by varying ␣ and  for a given value r and interpret the results in terms of the damping parameters b and c. III. EXPERIMENT The experimental setup shown in Fig. 1 is the same as described in Ref. 1 with slight modifications. A coil spring with force constant k = 30 N / m and unloaded length ⬃20 cm was connected vertically to a 50 N force sensor clamped on a metallic spindle rigidly anchored to the laboratory wall. The sensor has a resolution of 0.05 N and was connected through a Data Logger to a personal computer. The spring mass 共⬃0.05 kg兲 was much less than the container mass 共⬃1.5 kg兲 and can be neglected. The container was an inverted plastic soda bottle filled with sand and hooked from the bottom to the lower end of the spring. A thin wooden disk, 0.2 m in diameter and 4 mm thick, was mounted on the bottle neck to enhance the air drag. Specially designed aluminum funnel-shaped tips of different openings were used to vary the mass discharge rate. In all cases the mass discharge rate was constant and proportional to the outlet diameter to the 25 power.6 The bottle was set to oscillate vertically by pulling it down for about 0.2 m from its equilibrium position and gently released, simultaneously allowing the sand to pour out. The force sensor logged the instantaneous force versus time at a Digilov, Reiner, and Weizman
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Fig. 2. Plot of the net force versus time. The light line corresponds to the equilibrium position of a variable mass on a spring pendulum. The equation of this line is F0共t兲 = kz0共t兲 = m0g − rgt, where z0共t兲 is the stationary position of the spring 共see text兲. The slope of this equation rg = 0.0355 N / s and the coefficient m0g = 15.419 N are found by a linear fit to the chart record data F共t兲; r is the mass loss rate, and m0 is the initial mass of the container.
frequency of 10 s−1. The force data 共see Fig. 2兲 was imported to a spreadsheet for further analysis. The mass discharge rate r was determined from the slope of the line of F共t兲 and by directly measuring the weight loss rate at rest with the aid of the force sensor. The two methods give a slight difference. The F共t兲 data were converted to the equivalent displacement z共t兲 = F共t兲 / k about the stationary position, z0共t兲 = 0, with standard error ⬃2% and plotted as z共t兲 / A0 for various values of r 共see Fig. 3兲. The peaks of the oscillations give A共t兲 / A0, which is fitted to Eq. 共20兲 by varying ␣ and  共see Fig. 4兲. Typically, we used k = 30 N m−1, m0 ⬃ 1.5 kg, A0 ⬃ 0.2, so that b Ⰶ 1 kg s−1, r Ⰶ 1 kg s−1, and c Ⰶ 1 kg m−1. IV. RESULTS AND DISCUSSION We first analyze the damping caused by air drag. Figure 3共a兲 shows typical oscillations for r = 0. Although the decay appears to be exponential, a fit of A共t兲 / A0 to Eq. 共21兲 with ␣0 = 0.0035± 0.0001 s−1 and 0 = 4.59± 0.22 is much better than a fit to an exponential 关see Fig. 4共a兲兴. The damping parameters calculated from ␣0 and 0 using Table I are b0 = 0.0107± 0.0004 kg s−1 and c0 = 0.072± 0.005 kg m−1. Note that the effect of the air drag is apparent in both the parameters b and c. If only the linear drag term was important, then only b would be nonzero.
Fig. 4. Plots of the decay of the normalized amplitude versus time for different mass loss rates. Circles 共䊊兲: the peaks of the oscillations observed in Figs. 3共a兲–3共e兲. The solid line represents the fit of the data to Eq. 共21兲 for r = 0 and Eq. 共20兲 for r ⫽ 0. The bold curve in 共a兲–共c兲 is the fit of the data to an exponential.
Fig. 3. Plots of the normalized displacement as a function of time for different damping conditions. 共a兲 Typical behavior of the underdamped oscillations for a fixed mass on a spring pendulum caused by air drag alone. 共b兲–共e兲 The damping due to air drag for different values of the mass loss rate. 903
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Figure 3共b兲 shows the experimental results for r ⬃ 3.62 g s−1. Almost exponential decay is observed for short time, but for longer times, the exponential fit is not as good, especially for large amplitudes which decay more slowly Digilov, Reiner, and Weizman
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Table I. Comparison of damping parameters deduced from fitting the experimental data to Eqs. 共20兲 and 共21兲. D 共mm兲 m0 共kg兲 A0 共m兲 0 共s−1兲 r 共g s−1兲 ␣  b 共kg s−1兲 c 共kg m−1兲
0 1.544± 0.005 0.182± 0.002 4.41± 0.01 0 0.0035± 0.0001 4.59± 0.22 0.0107± 0.0004 0.072± 0.005 0
4.85 1.57± 0.005 0.197± 0.002 4.37± 0.01 3.62± 0.01 1.829± 0.063 3.098± 0.177 0.0151± 0.0006 0.056± 0.006 0.08
关see Fig. 4共b兲兴. The observed trend in the residuals is associated with the effect of the velocity-dependent drag force as is seen in the solid curve in Figs. 4共b兲 and 4共c兲, which is an excellent fit of the data to Eq. 共20兲. To see how the mass loss rate affects the damping parameters b and c, measurements for various r were made. Figures 3共c兲–3共e兲 show our results for r = 9.75, 18.2, and 36.9 g s−1. The values of b and c deduced from ␣ and  are shown in Table I and are plotted against r in Fig. 5. Although b and c are not determined with high accuracy for large values of r, it is clear that b increases approximately quadratically and c decreases linearly with r. For r → 24 g s−1, c → 0, and hence for r 艌 24 g s−1 only two damping mechanisms are important. An interesting feature suggested by our analysis of the data is that the sum A00c共r兲 + b共r兲 decreases approximately linearly with r 共see Fig. 6兲, so that at r 艌 52 g s−1, b and c are zero, that is, air drag is unimportant and energy loss due to the mass is dominant. The decay of the amplitude is described in this case by Eq. 共22兲. The measurements for r ⬇ 37 g s−1 关see Fig. 3共e兲兴 show a convex envelope. The fit to Eq. 共20兲 关see Fig. 4共e兲兴 yields ␣ ⬇  ⬇ 0, and A共t兲 = A0共1 − rt/m0兲
1/2
.
共23兲
As follows from Fig. 6, for r ⬇ 24 g s−1, b ⬇ 37 g s−1, Eq. 共20兲 becomes
Fig. 5. Damping parameters as a function of the mass loss rate determined from fitting the data to Eq. 共20兲 for different values of r. Diamond 共〫兲: quadratic damping parameter c. Circle 共䊊兲: linear damping parameter b. 904
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8 1.57± 0.005 0.205± 0.002 4.37± 0.01 9.75± 0.02 0.684± 0.049 2.531± 0.3 0.0182± 0.0013 0.044± 0.009 0.22
11 1.55± 0.005 0.192± 0.002 4.4± 0.01 18.2± 0.04 0.445± 0.104 0.696± 0.477 0.025± 0.006 0.016± 0.015 0.4
14 1.47± 0.005 0.187± 0.002 4.5± 0.01 36.9± 0.04 0 0 0.0185 0 0.8
A共t兲 = A0共1 − rt/m0兲.
共24兲
That is, the amplitude decays linearly with time, similar to the behavior in Fig. 3共d兲. The reasons for this behavior cannot be explained in terms of air drag for a fixed mass.7 For oscillatory motion of a body whose mass decreases at a constant rate, the drag force may depend on the viscosity and the density of the fluid, the characteristic dimension of the body d, the initial frequency 0, the initial amplitude A0, and the mass loss rate r. Dimensional analysis shows that the drag coefficient is a function of three dimensionless parameters: the Stokes number  = 0d2 / , the KeuleganCarpenter number KC= A / d 共or the oscillatory Reynolds number Re0 = A00d / = KC, the counterpart of steady motion兲, and the dimensionless mass loss rate = r / 0d3, that is, Fd = 共Re0,兲. A220d2
共25兲
A study of this dependence is a separate problem. Here, we present qualitative trends, observed at Re0 ⬇ 104: • For ⬍ 0.5, both the linear and quadratic damping mechanisms are operative with b increasing and c decreasing with r 共see Fig. 5兲. In this case, the time dependence of the envelope is concave as described by Eq. 共20兲.
Fig. 6. The quantity A00c共r兲 + b共r兲 as a function of the mass loss rate. Diamond 共〫兲: data in Table I. The straight line is the linear approximation to these data. Digilov, Reiner, and Weizman
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• For 0.5艋 ⬍ 1, the quadratic damping parameter vanishes and b decreases with r; ⬇ 0.5 corresponds to linear decay in the amplitude as described by Eq. 共24兲; for ⬎ 0.5 the envelope is convex, a special case of which is Eq. 共23兲. • For 艌 1, both b and c are negligible and the decay of A共t兲 due to the mass loss alone is given by Eq. 共22兲. V. CONCLUSION We have derived approximate results for damped harmonic motion with a variable mass. By modeling the damping force, students can find the corresponding form of the damped oscillations. For instance, the oscillatory behavior of a sliding variable-mass-spring system with a linear damping term due to air drag and a velocity-independent dry friction 共Coulomb friction兲 could be analyzed. Another model that could be tested experimentally is a variable mass vibration system with a damping force proportional to 兩v兩n, for arbitrary n. The experimental verification of a model would provide students with experience in dimensional analysis, data collection, and data analysis.
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ACKNOWLEDGMENTS Support from the Israel Ministry of Absorption and Immigration is gratefully acknowledged.
a兲
Author to whom correspondence should be addressed. Electronic mail:
[email protected] 1 J. Flores, G. Solovey, and S. Gil, “Variable mass oscillator,” Am. J. Phys. 71, 721–725 共2003兲. 2 V. K. Gupta, G. Shanker, and N. K. Sharma, “Experiment on fluid drag and viscosity with an oscillating sphere,” Am. J. Phys. 54, 619–622 共1986兲. 3 P. T. Squire, “Pendulum damping,” Am. J. Phys. 54, 984–991 共1986兲. 4 X. Wang, C. Schmitt, and M. Payne, “Oscillations with three damping effects,” Eur. J. Phys. 23, 155–164 共2002兲. 5 J. Copeland, “Work-energy theorem for variable mass system,” Am. J. Phys. 50, 599–601 共1982兲. 6 M. Yersel, “The flow of sand,” Phys. Teach. 38, 290–291 共2000兲. 7 L. D. Landau and E. M. Lifshitz, Fluid Mechanics 共Pergamon, Oxford, 1959兲, pp. 49, 91–92, 95–96.
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