Effective Torsion and SpringConstants in a Hybrid Translational-Rotational OscillatorZein Nakhoda1 and Ken Taylor, Lake Highlands High School, Dallas, TX (Richardson ISD)

Atorsion oscillator is a vibrating system that experi-ences a restoring torque given by T = -",e when itexperiences a rotational displacement e from its

equilibrium position. The torsion constant", (kappa) isanalogous to the spring constant k for the traditional trans-lational oscillator (for which the restoring force F is propor-tional to the linear displacement x of the mass). An effectivetorsion oscillator can be constructed by integrating a spring'stranslational harmonic properties into an Atwood? arrange-ment where a disk serves as the pulley for the system and thespringes) exert restoring torques on the oscillating disk. Botheffective torsion constants and effective spring constants canbe expressed in terms of adjustable parameters of the system.These expressions enable one to theoretically describe themotion of the hybrid oscillator and to calculate its period.A comparison of the translational and rotational interpreta-tions teaches of their analogous mathematical propertiesand challenges the intuitive skills of those considering suchsystems.

Torsion oscillator with two springsTo construct the oscillator, two fixed springs are positioned

to exert opposing torques on the disk (see Fig. 1). The adjust-able parameters of the system include the force constants of thesprings, the radii on which the springs exert restoring torqueson the disk, and the moment of inertia of the disk. PASCO'sintroductory rotational apparatus.' is used for the rotationalpart of the oscillator, and two springs are attached horizon-tally by strings perpendicular to the radii on which they exertrestoring forces. The fixed ends of the springs are attached toseparate PASCO force sensors to record the force exerted byeach spring. A PASCO rotary motion sensor positioned on theaxis of the main disk records the angular position.

Applying Hooke's law, with the measured force constantsk 1 and k2 of the two springs, an expression for the effective

Force sensor

Fig. 1. Double-spring torsion oscillator arrangement.

106 THE PHYSICS TEACHER. Vol. 49, FEBRUARY 2011

torsion constant of the system may be obtained. If radii R 1 andR2 are taken as the moment arms to which restoring springforces Tl and T2 are applied, then

T = -"'effOT.,RI - T1Rl = -"'effO

-k,x,R, + k1x1R1 = -"'effOand

k,x,R,

(1)

This then gives

"'eff = k,R,l + klR12

for the effective torsion constant "'eff of the system. (Thesubstitution for the X2 term comes from the relation

B=~=-~.)R, R2

Ignoring damping, the differential equation proposed forapproximating the behavior of the system as a torsion oscilla-tor may be expressed as

le+(k,R,2+k2R/)B=O, (3)

where I is the moment of inertia of the disk. Pursuant to thestandard form of this equation, the angular frequency of theoscillations may be found by inspection to be the square rootof the ratio of the Hooke's law coefficient to the accelerationcoefficient. For the period of the motion, this then leads tothe relation

(2)

T = 21f1

(4)2 2 .k.R, +k2R2

As a tool for teaching the analogous properties of rota-tional and linear oscillating systems, one can conceptualizethe inertia of the disk as a nonrotating mass moving back andforth between the two springs. The system can be describedtranslationally by manipulating Eq. (3) such that

(tmDRD2)[:D 1+ (k,R,2 + k2R/)[ ;D 1= °and

(5)

001: '0.'119/1.3543587

Table I. Double-spring oscillator data.

Config.1 2 3

k1 26.5 N/m 26.5 N/m 26.5 N/m

k2 26.5 N/m 5.69 N/m 5.69 N/m

R1 0.127 m 0.127 m 0.025 m

R2 0.127 m 0.127 m 0.127 m

AverageTexperimental 0.567 s 0.721 s 1.578 s

Ttheoretical 0.589 s 0.755 s 1.653 s

0.151 ~ 1 1 1

f~s!~

01 1 x,_·-.--, -Y'" ...·--1 1

-0.30' T I I I-0.50 -0.25 0 0.25 0.50

Angular Position (rad)

Fig. 2. Representative data showing correspondence of theoreti-cal and experimental values for the effective torsion constant.Data were recorded for many oscillations, causing the appear-ance of multiple overlapping plots.

Equation (5) describes the system as a translational oscilla-tor, where mO,eff and Ro are, respectively, the effective massand radius of the disk. Since the moment of inertia of thedisk is given as 1 = 1moRo 2

, we see from inspection that_ 1 4

mO,efT - '2 mO'

As Eq. (5) reveals, the effective spring constant for thesystem, keff, takes the form (k1R1

2 + k2R22)/Ro2, providing alinear spring constant for the rotational system. This, in turn,leads to a period expression for the translational view thattakes the form

T ~ 2, I(k,R,7i~'R'T (6)

Interestingly, ifboth springs were to exert their forces atthe perimeter of the disk, there would be a uniform lineardisplacement, and the effective spring constant for the systemoscillating translationally would reduce to keff = k I + k2. Thisis consistent with the behavior of two springs in parallel. Moregenerally, the two springs may still be considered as acting in

parallel but with individual effective spring constants of

k, [ ~ rand k, [ ~ rIt should be noted that no particular form of damping was

assumed for the two differential equations above [viz., Eq. (3)and Eq. (5)). Careful review of the decaying oscillations reovealed a complex presence of both air resistance and dry fric-tion as noted by the respective presence of both exponentialand linear decay.

Data for this paper were acquired for nine separate experi-ments using three different configurations of springs andradii (see Table I). The periods of oscillation were measuredand compared to theoretical predictions using the derived ef-fective torsion and effective spring constants seen in Eq. (2)and Eq. (5). (The radii RI, R2, and Ro used in these equationswere measured directly while the values for k I and k2 were ob-tained in separate experiments not described here.)

In Table I, the experimental periods are averages of thethree trials done for each configuration. In all cases, thetranslational and rotational predictions for the period werethe same, rounded to three decimal places and predicted theexperimental period with an average 4.28% error. The theo-retical period was greater than the experimental in all cases.Some discrepancies can be attributed to errors in measuringthe spring constants of the individual springs, error in thegiven moment of inertia of the disk and consequent effectivemass, and the effect of the springs' masses in the system. Inaddition, damping of the motion should be considered as anexpected source. No extensive effort was made to isolate thesources of error. 5

Experimental values for the effective torsion constant andeffective spring constant can be found by graphing Tnet =-"'effB using TIRI - T2R2 = - "'effB with data from the forcesensors and rotary motion sensor (see Fig. 2). Once "'eff isfound, the relationship

k "'efTcff = R 2 [see Eqs. (2) and (5))

o

can be used to find the effective spring constant.

Torsion oscillator with one spring and asuspended mass

A similar oscillating system can be created using only onespring if the other spring is replaced by a suspended massexerting a torque on the disk (see Fig. 3). The effective springand torsion constants of this hybrid oscillator can again beexpressed using known parameters including the stiffness ofthe spring, the amount of hanging mass, the radii on whichthe spring and weight of the mass exert their forces, and themoment of inertia of the disk.

As before, the spring is attached horizontally to the diskwith a string that exerts force Ts perpendicularly to the radiuson which the torque is exerted. The mass is suspended be-

THE PHYSICS TEACHER. Vol. 49, FEBRUARY 2011 107

m

p~. ~~~cio I k Force Sensor

Fig. 3. Side view of single-spring hybrid oscillator.

I k Force Sensor

PowerAmplifier

Fig. 4. Single-spring magnetically driven oscillator arrangement.

neath a light pulley with a string that exerts force Tm tangen-tially to the disk. The far end of the spring is fixed to a forcesensor, and a rotary motion sensor is attached to the axis ofthe disk to record angular position.

By combining the translational and rotational aspects ofthe system, the differential equation describing the motionof the hybrid oscillator can be derived. Hence, setting upNewton's law for the mass and disk and solving the equationssimultaneously leads to

ma = mg - Tm; T.nRm+ 7;,Rs = Ie

ma =mg- Tm; TmRm-kxRs = Ie

Tm= mg - mI\ne = Ie ~ kxRsm

.. Ie R,mg - mRrrf)= - +kx-Rm Rm

[I ) .. R2-+mRm e+k-s e = mgs; s;

and finally

(I +mRm2) e +(kRs 2) e =mgRm .

In effect, the hanging mass m contributes to the momentof inertia of the disk, I, in the form mRm 2, where Rm is theradius on which the hanging mass exerts its torque. The effec-tive torsion constant, "'eff, is expressed as kRs2, where k is theforce constant of the spring and Rs is the radius on which thespring exerts its torque.

Just as the weight of a mass suspended on a vertical springhas no influence on the period of the vibrations of the sys-tern," the same situation presents itself here. Hence, the torquecomponent mgRm caused by the weight of the mass has no in-fluence on the period of the motion. Thus, by inspection onceagain, the period of the oscillations can be expressed as

I +mRm2T = 27f kR 2 (8)

s

108 THE PHYSICS TEACHER. Vol. 49, FEBRUARY 2011

0.45

~ \~

l!,1\'/ -,

1.// ,1,\/lot

if""" ~ r--

0.30

~~I

0.15

o1.5 1.61.1 1.2 1.3 1.4

Frequency (Hz)

Fig. 5. Resonance curves for moderate and light damping.

For pedagogical purposes, we can once again conceptual-ize the disk as an additional, translating mass connected to thespring and the system can be interpreted translation ally usingthe following differential equation:

[mD,eff mRm:)x [kRs:)x mgRm2. (9)RD RD RD

The effective spring constant keff is expressed as kR/IRo2,where Ro is the radius of the disk. Notably, if the spring exertsits restoring force at the disk's perimeter, the effective springconstant reduces to k. As before, this leads to a period expres-sion for the translational view that takes the form

T = 27f

(~R::)R 2

mD,eif+m Rm2D

(10)

(7)

Data for this arrangement were taken from 11 trials usingfour different configurations of springs and radii, and wererecorded similarly to the double spring data. The hangingmass varied for each trial for each of the four spring and radiiconfigurations. The periods of oscillation were measured andcompared to theoretical predictions using the derived effectivetorsion and effective spring constants. In all cases, the transla-tional and rotational predictions for the period were the same,rounded to three decimal places except for one, which differedby one thousandth. The theory predicted the experimentalperiod with an average 4.46% error. As before, the theoreticalperiod was greater than the experimental in all cases.

Magnetically driven torsion oscillatorTo further study the single spring hanging mass system, a

driven arrangement was set up to study its resonance quali-ties. The system mass was magnetically driven by suspendinga linear arrangement of magnets beneath the mass hanger andabove a solenoid. The solenoid was energized by a PASC03

750 Interface and Power Amplifier II to serve as an electro-

magnet acting on the magnets? (see Fig. 4). The differentialequation describing the motion of the system is the same asEq. (7) with the exception of the inclusion of the driving forceF(t). Hence,

(I +mRm2)e +(kRs2)() = mgRm +F(t). (11)

Since the field acting on the magnet is not uniform andthe magnet is dipolar, the actual force function between themagnet and electromagnet is not known. Hence, F(t) cannotbe specified for Eq. (11). A specific waveform can, of course,be selected at the interface in order to give the coil current(and magnetic field) a specific form. For this experiment, asine wave of amplitude 2.2 V and variable frequency was used.Again, due to light damping, it was not included in the equa-tion.

By driving the system at varying frequencies, a resonancecurve was obtained that describes the frequency responseof the system (see Fig. 5). Both a moderately damped and alightly damped case were studied. In all cases, only the steady-state amplitudes of the driven oscillations were utilized for thegraph.

When setting up the magnetic driver, it greatly facilitatesselections of driving amplitudes if the magnet is able to travelin and out of the solenoid. Neodymium magnets were usedin the experiment because of their relative strength and smallsize.

ConclusionsThis paper has sought to demonstrate several things to

readers: (1) a method for calculating the effective torsionconstant of a hybrid oscillator with adjustable parameters ofthe system, (2) the use of hybrid oscillators for teaching theconcept of analogy and modeling (e.g., rotational and linearquantities), and (3) the use of an electromagnetic system forconveniently driving oscillations.

With the exception of the resonance experiment, the labexercises described may be easily implemented at both thehigh school and college level. If need be, teacher modificationcan adapt the experiment to local apparatus and time con-straints. If care is used, the resonance experiment can be doneover a long college lab or two to three days of high school ef-fort.

References1. Zein Nakhoda (currently attending Swarthmore College) wrote

this paper upon graduating from Lake Highlands High School.He is the product of two years of study in high school physics.With the exception of the coauthor's suggestion of writing apaper on the properties of a driven and nondriven hybrid oscil-lating system, the experimental procedure, data analysis, math-ematical derivations, and major aspects ofthe text are his own.The working relationship has been similar to that of a graduatestudent and supervisor.

2. Thomas B. Greenslade [r., "Atwood's machine;' Phys. Teach. 23,24-28 (Jan. 1985).

3. PASCO, Roseville, CA; www.pasco.com.4. The moment of inertia for a single particle, I = mR2, was

used to derive the effective mass from the inertia of a soliddisk:1 = moerrRo2

tmoRo2 = -;»:tmoRo2 = mo.errRo 2

5. Table I: For Average Texperimental, there was additional errordue to the rotary motion sensor, but no estimate was available.Values for R 1 and R2 were supplied by PASCO; errors are notknown.

6. Paul A. Tipler, Physics for Scientists and Engineers, 1999, 4th ed.(W.H. Freeman and Company, New York), p. 4l3.

7. Ken Taylor, "Resonance effects in magnetically driven mass-spring oscillations;' Phys. Teach. 49,49-50 (Jan. 2011).

Zein Nakhoda is a graduate of Lake Highlands High School in Dallas,TX (Richardson ISO) and currently attends Swarthmore College inPennsylvania.Swarthmore College,Swarthmore, PA19081;wnakhod1@swarthmore.edu

KenTaylor teaches AP Physics 8 and Cat Lake Highlands High School inDallas, TX (Richardson ISO).

Lake Highlands HighSchool (RichardsonISO),9449Church Road,Dallas,TX75238; ken.taylor@risd.org

TEL-AtomicIncorporated

• Short oscillation periods of 2-4 minutes

• Complete experimental results in a single lab period

• No more optical lever jitters due to revolutionarysensor design

• 24 bit resolution Computerized Cavendish

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