Effective Torsion and Spring
Constants in a Hybrid TranslationalRotational Oscillator
Zein Nakhoda1 and Ken Taylor, Lake
Highlands High School, Dallas, TX (Richardson ISD)
A
torsion oscillator is a vibrating system that experiences a restoring torque given by T =
when it
experiences a rotational displacement
from its
equilibrium position. The torsion constant", (kappa) is
analogous to the spring constant k for the traditional translational oscillator (for which the restoring force F is proportional to the linear displacement x of the mass). An effective
torsion oscillator can be constructed by integrating a spring's
translational harmonic properties into an Atwood? arrangement where a disk serves as the pulley for the system and the
springes) exert restoring torques on the oscillating disk. Both
effective torsion constants and effective spring constants can
be expressed in terms of adjustable parameters of the system.
These expressions enable one to theoretically describe the
motion of the hybrid oscillator and to calculate its period.
A comparison of the translational and rotational interpretations teaches of their analogous mathematical properties
and challenges the intuitive skills of those considering such
systems.
-",e
e
Torsion oscillator with two springs
To construct the oscillator, two fixed springs are positioned
to exert opposing torques on the disk (see Fig. 1). The adjustable parameters of the system include the force constants of the
springs, the radii on which the springs exert restoring torques
on the disk, and the moment of inertia of the disk. PASCO's
introductory rotational apparatus.' is used for the rotational
part of the oscillator, and two springs are attached horizontally by strings perpendicular to the radii on which they exert
restoring forces. The fixed ends of the springs are attached to
separate PASCO force sensors to record the force exerted by
each spring. A PASCO rotary motion sensor positioned on the
axis of the main disk records the angular position.
Applying Hooke's law, with the measured force constants
k 1 and k2 of the two springs, an expression for the effective
torsion constant of the system may be obtained. If radii R 1 and
R2 are taken as the moment arms to which restoring spring
forces Tl and T2 are applied, then
T
= -"'effO
= -"'effO
+ k1x1R1 = -"'effO
T.,RI - T1Rl
-k,x,R,
and
k,x,R,
This then gives
= k,R,l + klR12
"'eff
B=~=-~.)
R,
R2
Ignoring damping, the differential equation proposed for
approximating the behavior of the system as a torsion oscillator may be expressed as
le+(k,R,2+k2R/)B=O,
T
=
1
21f
2
k.R, +k2R2
THE PHYSICS
1+ (k,R,2 + k2R/)[
and
Vol.
49, FEBRUARY 2011
;D 1 = °
(5)
arrangement.
TEACHER.
(4)
2 .
As a tool for teaching the analogous properties of rotational and linear oscillating systems, one can conceptualize
the inertia of the disk as a nonrotating mass moving back and
forth between the two springs. The system can be described
translationally by manipulating Eq. (3) such that
Force sensor
106
(3)
where I is the moment of inertia of the disk. Pursuant to the
standard form of this equation, the angular frequency of the
oscillations may be found by inspection to be the square root
of the ratio of the Hooke's law coefficient to the acceleration
coefficient. For the period of the motion, this then leads to
the relation
2
torsion oscillator
(2)
for the effective torsion constant "'eff of the system. (The
substitution for the X2 term comes from the relation
(tmDRD )[:D
Fig. 1. Double-spring
(1)
001: '0.'119/1.3543587
Table I. Double-spring
oscillator
data.
Config.1
2
3
k, [ ~
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
0.567 s
0.721 s
1.578 s
0.589 s
0.755 s
1.653 s
~
1
Average
Texperimental
Ttheoretical
0.151
f
~
s
!
01
1
-0.30'
-0.50
T
-0.25
x,_·-.--,
1
-Y'" ...
1
·--1
1
~
I
0
I
0.25
I
0.50
Angular Position (rad)
Fig. 2. Representative data showing correspondence of theoretical and experimental values for the effective torsion constant.
Data were recorded for many oscillations, causing the appearance of multiple overlapping plots.
Equation (5) describes the system as a translational oscillator, where mO,eff and Ro are, respectively, the effective mass
and radius of the disk. Since the moment of inertia of the
disk is given as 1 = moRo 2, we see from inspection that
1
_
mO,efT -
1
rand
k, [ ~
It 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 reo
vealed a complex presence of both air resistance and dry friction as noted by the respective presence of both exponential
and linear decay.
Data for this paper were acquired for nine separate experiments using three different configurations of springs and
radii (see Table I). The periods of oscillation were measured
and compared to theoretical predictions using the derived effective torsion and effective spring constants seen in Eq. (2)
and Eq. (5). (The radii RI, R2, and Ro used in these equations
were measured directly while the values for k I and k2 were obtained in separate experiments not described here.)
In Table I, the experimental periods are averages of the
three trials done for each configuration. In all cases, the
translational and rotational predictions for the period were
the same, rounded to three decimal places and predicted the
experimental period with an average 4.28% error. The theoretical period was greater than the experimental in all cases.
Some discrepancies can be attributed to errors in measuring
the spring constants of the individual springs, error in the
given moment of inertia of the disk and consequent effective
mass, and the effect of the springs' masses in the system. In
addition, damping of the motion should be considered as an
expected source. No extensive effort was made to isolate the
sources of error. 5
Experimental values for the effective torsion constant and
effective spring constant can be found by graphing T net =
-"'effB using TIRI - T2R2 = - "'effB with data from the force
sensors and rotary motion sensor (see Fig. 2). Once "'eff is
found, the relationship
4
'2 mO'
k
"'efT
cff
As Eq. (5) reveals, the effective spring constant for the
system, keff, takes the form (k1R12 + k2R22)/Ro2, providing a
linear spring constant for the rotational system. This, in turn,
leads to a period expression for the translational view that
takes the form
T ~ 2, I
r
parallel but with individual effective spring constants of
(k,R,7i~'R'T
(6)
Interestingly, ifboth springs were to exert their forces at
the perimeter of the disk, there would be a uniform linear
displacement, and the effective spring constant for the system
oscillating translationally would reduce to keff = k I + k2. This
is consistent with the behavior of two springs in parallel. More
generally, the two springs may still be considered as acting in
=R
2
[see Eqs. (2) and (5))
o
can be used to find the effective spring constant.
Torsion oscillator with one spring and a
suspended mass
A similar oscillating system can be created using only one
spring if the other spring is replaced by a suspended mass
exerting a torque on the disk (see Fig. 3). The effective spring
and torsion constants of this hybrid oscillator can again be
expressed using known parameters including the stiffness of
the spring, the amount of hanging mass, the radii on which
the spring and weight of the mass exert their forces, and the
moment of inertia of the disk.
As before, the spring is attached horizontally to the disk
with a string that exerts force Ts perpendicularly to the radius
on which the torque is exerted. The mass is suspended be-
THE PHYSICS
TEACHER.
Vol.
49, FEBRUARY 2011
107
p~.
cio
0.45
~~~
I
k
Force Sensor
m
0.30
Fig. 3. Side view of single-spring
~
~
hybrid oscillator.
k
I
I
~\
~
l!,1\
0.15
'/
Force Sensor
-,
,1,\
1.//
/lot
o
1.1
r--
~
if"""
Power
Amplifier
1.2
1.3
1.4
1.5
1.6
Frequency (Hz)
Fig. 4. Single-spring
magnetically
driven oscillator
arrangement.
neath a light pulley with a string that exerts force Tm tangentially to the disk. The far end of the spring is fixed to a force
sensor, and a rotary motion sensor is attached to the axis of
the disk to record angular position.
By combining the translational and rotational aspects of
the system, the differential equation describing the motion
of the hybrid oscillator can be derived. Hence, setting up
Newton's law for the mass and disk and solving the equations
simultaneously leads to
ma
= mg
ma = mg-
- Tm; T.nRm+ 7;,Rs
Tm; TmRm-kxRs
Tm = mg - mI\ne
=
=
Ie
= Ie
Fig. 5. Resonance
[mD,eff
R
T
= mg
and finally
(I
+ mRm2) e +(kRs 2)
e = mgRm
.
(7)
In effect, the hanging mass m contributes to the moment
of inertia of the disk, I, in the form mRm 2, where Rm is the
radius on which the hanging mass exerts its torque. The effective torsion constant, "'eff, is expressed as kRs2, where k is the
force constant of the spring and Rs is the radius on which the
spring exerts its torque.
Just as the weight of a mass suspended on a vertical spring
has no influence on the period of the vibrations of the systern," the same situation presents itself here. Hence, the torque
component mgRm caused by the weight of the mass has no influence on the period of the motion. Thus, by inspection once
again, the period of the oscillations can be expressed as
T
= 27f
I +mRm2
kR 2
THE PHYSICS
mgRm2.
RD
(9)
2
(8)
TEACHER.
= 27f
D
(~R::)
(10)
Data for this arrangement were taken from 11 trials using
four different configurations of springs and radii, and were
recorded similarly to the double spring data. The hanging
mass varied for each trial for each of the four spring and radii
configurations. The periods of oscillation were measured and
compared to theoretical predictions using the derived effective
torsion and effective spring constants. In all cases, the translational and rotational predictions for the period were the same,
rounded to three decimal places except for one, which differed
by one thousandth. The theory predicted the experimental
period with an average 4.46% error. As before, the theoretical
period was greater than the experimental in all cases.
Magnetically driven torsion oscillator
To further study the single spring hanging mass system, a
driven arrangement was set up to study its resonance qualities. The system mass was magnetically driven by suspending
a linear arrangement of magnets beneath the mass hanger and
above a solenoid. The solenoid was energized by a PASC03
750 Interface and Power Amplifier II to serve as an electro-
s
108
[kRs:)x
RD
mD,eif+m Rm2
.. Ie
R,
mg - mRrrf) = - + kxRm
Rm
s;
mRm:)x
RD
The effective spring constant keff is expressed as kR/IRo2,
where Ro is the radius of the disk. Notably, if the spring exerts
its restoring force at the disk's perimeter, the effective spring
constant reduces to k. As before, this leads to a period expression for the translational view that takes the form
Ie ~ kxRs
) ..
R2
e+k-s
e
and light damping.
For pedagogical purposes, we can once again conceptualize the disk as an additional, translating mass connected to the
spring and the system can be interpreted translation ally using
the following differential equation:
m
I
[-+mRm
s;
curves for moderate
Vol. 49, FEBRUARY 2011
magnet acting on the magnets? (see Fig. 4). The differential
equation describing the motion of the system is the same as
Eq. (7) with the exception of the inclusion of the driving force
F(t). Hence,
(I mRm2
kRs2)() = mgRm F(t).
(11)
+
)e + (
3.
4.
PASCO, Roseville, CA; www.pasco.com.
The moment of inertia for a single particle, I = mR2, was
used to derive the effective mass from the inertia of a solid
disk:
1 = moerrRo2
+
Since the field acting on the magnet is not uniform and
the magnet is dipolar, the actual force function between the
magnet and electromagnet is not known. Hence, F(t) cannot
be 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, a
sine wave of amplitude 2.2 V and variable frequency was used.
Again, due to light damping, it was not included in the equation.
By driving the system at varying frequencies, a resonance
curve was obtained that describes the frequency response
of the system (see Fig. 5). Both a moderately damped and a
lightly damped case were studied. In all cases, only the steadystate amplitudes of the driven oscillations were utilized for the
graph.
When setting up the magnetic driver, it greatly facilitates
selections of driving amplitudes if the magnet is able to travel
in and out of the solenoid. Neodymium magnets were used
in the experiment because of their relative strength and small
size.
tmoRo2
t moRo
2
-;»:
=
= mo.errRo
2
5.
Table I: For Average Texperimental, there was additional error
due to the rotary motion sensor, but no estimate was available.
Values for R 1 and R2 were supplied by PASCO; errors are not
known.
6.
Paul A. Tipler, Physics for Scientists and Engineers, 1999, 4th ed.
(W.H. Freeman and Company, New York), p. 4l3.
Ken Taylor, "Resonance effects in magnetically driven massspring oscillations;' Phys. Teach. 49,49-50 (Jan. 2011).
7.
Zein Nakhoda is a graduate of Lake Highlands High School in Dallas,
TX (Richardson ISO) and currently attends Swarthmore
Pennsylvania.
College in
Swarthmore College,Swarthmore, PA19081;wnakhod1@
swarthmore.edu
Ken Taylor teaches AP Physics 8 and C at Lake Highlands High School in
Dallas, TX (Richardson ISO).
Lake Highlands High School (Richardson ISO),9449
Church Road,Dallas,TX 75238; [email protected]
Conclusions
This paper has sought to demonstrate several things to
readers: (1) a method for calculating the effective torsion
constant of a hybrid oscillator with adjustable parameters of
the system, (2) the use of hybrid oscillators for teaching the
concept of analogy and modeling (e.g., rotational and linear
quantities), and (3) the use of an electromagnetic system for
conveniently driving oscillations.
With the exception of the resonance experiment, the lab
exercises described may be easily implemented at both the
high school and college level. If need be, teacher modification
can adapt the experiment to local apparatus and time constraints. If care is used, the resonance experiment can be done
over a long college lab or two to three days of high school effort.
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References
1.
2.
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 a
paper on the properties of a driven and nondriven hybrid oscillating system, the experimental procedure, data analysis, mathematical derivations, and major aspects ofthe text are his own.
The working relationship has been similar to that of a graduate
student and supervisor.
Thomas B. Greenslade [r., "Atwood's machine;' Phys. Teach. 23,
24-28 (Jan. 1985).
THE PHYSICS
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49.
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2011
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