N E W A P P ROAC H E S
weights of those components, such as the tail itself
and the linkage members, which are also raised in
the process and to the force required to
counterbalance the spring loading within the tail
which returns the device to its original state after
use. All of these factors give rise in practice to some
loss of efficiency. Further measurements could be
made with the spring balance attached at H and K
in turn to investigate the changes in mechanical
advantage which result when the number of
elements
in
the
linkage
is
reduced.
If the commercial device is not available then the
linkage, which is the essential component for
teaching purposes, could be very easily improvised
from metal strips such as Meccano. In this case the
effects of the spring loading in the tail and the
weight of the tail of the fish will be eliminated.
Through this improvisation it also becomes possible
to increase the number of elements beyond four and
explore the eventual limits to the development of the
machine.
Note. The Lazyfish is manufactured and distributed
by by Bacchanal Limited, Unit 15, Hale Trading
Estate, Lower Church Lane, Tipton, West Midlands
DY4 7PQ (tel: 0121 520 4727; fax: 0121 520 7637).
Received 22 April 1996, in final form 10 July 1996
Reference
Sandor B I 1983 Engineering Mechanics: Statics
and Dynamics (Englewood Cliffs, NJ: Prentice-Hall)
Speed of sound in tuning fork metal
V Anantha Narayanan Savannah State College, GA, USA and
Radha Narayanan Student, Windsor Forest High School,
Savannah, GA, USA
A procedure to find the speed of sound in tuning
fork metal is described. The formula needed is
extracted from the literature and explained. Since
the equipment needed for this project is readily
available in most high school and introductory
level college science laboratories, this exercise can
be done without any additional cost.
The details of the tuning fork construction formula
are extracted from the literature and presented here.
Calculation procedures to evaluate the speed of
sound in the tuning fork material are outlined.
Theory
The vibrations of the tuning fork are controlled by
the elasticity and the inertia of the prongs of the
fork. The complex vibrations of the tuning fork
involve bending deformation of the prongs.
Transverse motions of the prongs cause an up and
down motion in the stem of the tuning fork [1, 2].
Tuning forks are commonly used in resonance in
experiments with air columns to determine the
speed of sound in air very accurately. The frequency
of a tuning fork can be determined experimentally
by using a sonometer. Both these experiments are
frequently performed in introductory level physics
courses in high school and/or college. An
elementary derivation of the tuning fork frequency
formula using dimensional analysis is also given in a
book [3].
The mathematical treatment of transverse vibrations
in a straight rod or bar is very complicated [1, 4].
The tuning fork is an example of a bar clamped at
one end. The symmetrical modes of a tuning fork are
treated as being due to two vibrating bars fixed at
their lower ends. For such a bar the vibration nodes
389
N E W A P P ROAC H E S
for overtones are not evenly spaced and therefore
the overtones are not integral multiples of the
fundamental tone. The mathematical solution and
the essential details of the transverse vibrations of a
bar clamped at one end are succinctly summarized in
references [1, 4–9]. Such analysis leads to the
following formula for the frequency:
or
p EK 2 1/2
fn 5}}2 }} (1.1942,2.9882, 52, 72,.. . (2n21)2). (1)
8L
r
where l 5 6.173L2/t is the wavelength in m of the
compression waves in the metal for the fundamental
frequency f1.
1 2
This expression is the result of applying the
boundary conditions to the vibrations of a finite bar,
thereby limiting the allowed modes to a discrete set
of frequencies. Here f 5 frequency in Hz, n 5 1, 2,
3, 4, . . ., E 5 Young’s modulus of the material of the
bar in N m22, r 5 density of the bar metal in kg
m23, L 5 length of the bar in m, and K 5 radius of
gyration of the bar (of rectangular cross section with
t 5 thickness of the bar in m in the direction of
vibration) = t/121/2.
Figure 1 gives a pictorial representation of t and L
for a tuning fork. In equation (1) the numerical
terms (1.194)2 for f1 and (2.988)2 for f2 should be
used as given by detailed derivations in the original
treatises dealing with the theory without rounding
them off [1, p 73]. They are numerically significant
and result from the trigonometric solutions involved
in the derivations. If we substitute v 5 speed of
sound in the bar in m s21 5 (E/r)1/2, we can write
f1 5 pvK 3 1.1942/8L2
(2)
f2 5 pvK 3 2.9882/8L2
(3)
f3 5 pvK 3 52/8L2
(4)
f4 5 pvK 3 72/8L2.
(5)
v 5 f1L2/0.162t 5 6.173f1L2/t.
We can rewrite equation (7) as
v 5 lf1
An experiment involving the formula of a tuning fork
as f1 = constant 3 t/L2 is given as a laboratory
exercise in a manual [10]. However, it does not give
the details about the relation of this constant to v, l
and the moment of inertia or the connecting
references needed to appreciate a determination of
v by experiment.
We can rearrange equation (2) as
f1 5 (p/8) (v/L2) (t/121/2) 1.1942
390
(6)
(8)
When a tuning fork is gently struck the amplitude of
the overtones is not entirely negligible. Thus,
immediately after being struck, a distinct metallic
quality to the sound is produced and lasts for a short
time. The amplitudes of the overtones are much
smaller than the fundamental. The high-frequency
metallic sound rapidly dies out [1, 4, 7, 8]. The final
sustaining sound is a mellowed pure tone of the
fundamental mode. This is the frequency number
that is printed on laboratory tuning forks by the
manufacturers.
Also f2 /f1 5 6.26, f3 /f1 5 17.54 and f4 /f1 5
34.37. Hence the first four frequencies are f1, 6.26f1,
17.54f1 and 34.37f1. The extreme anharmonicity of
the vibrations of the tuning fork is evident from
these frequency values, in contrast to the integer
multiples of f as overtone frequencies in a vibrating
string.
5 0.162vt/L2 5 5 5 5 5 5
(7)
Figure 1. Tuning fork: representation of L and t.
N E W A P P ROAC H E S
Experimental details
Besides a collection of tuning forks, the items
needed to perform this experiment are a metre rule
and a vernier caliper.
In table 1 are listed the f1 values of nine tuning forks
made of an aluminium alloy as printed by the
manufacturer, and the measured values of t and L in
columns 3 and 4. All these tuning forks happen to
have the same t values. Column 5 gives the values
of calculated wavelength. Column 6 gives the
values of v calculated from equation (8).
Results
The speed of sound in aluminium can be calculated
using E 5 7 3 1010 N m22 and r 5 2700 kg m23
and the formula v 5 (E/r)1/2. We take 51 3 102 m
s21 as the table value of v. The average of the
experimental value of v from the last column in
table 1 is 49 3 102 m s21. The apparent
percentage error in the value of v is about 4%
from these data.
Taking logarithms of equation (3), then
differentiating and finally multiplying by 100
throughout, we get the percentage error
Dv
Df
2DL
}} 3 100% 5 }}1 3 100% 1 }} 3 100%
v
f1
L
2Dt
1 }} 3 100% .
t
(9)
The manufacturer of the tuning forks specifies the
right-hand side of equation (9), which is an error in
f1, to be 6 0.5%. These frequencies as given by the
manufacturer seem quite reliable on the basis of our
experiments involving these tuning forks in
resonance in air columns and the sonometer. We
estimate DL to be no more than 0.001 m, and the
term (2DL/L) 3 100% varies from 1.3% for fork #1
to 1.9% for fork #9, giving an average of about
1.6% error due to this term. The Dt in our
measurement is about 0.0002 m, so the error term
(Dt/t) 3 100% may amount to 1.4%. If we assume
that all these errors add cumulatively, the
percentage error in the extreme case for the
calculated value of v can be 3.5%.
Another contributing factor can be due to the fact
that the speed of sound in the aluminium alloy used
to construct the fork may not be exactly 51 3 102 m
s21. Pure aluminium is soft and lacks strength. It is
alloyed with very small amounts of other elements
like copper, magnesium and silicon. Three common
commercial aluminium alloys are listed in the
Handbook of Physics and Chemistry [11]. Using the
values of E and r listed therein the v were calculated
to be 50.3, 50.8 and 51.3 3 102 m s21. Thus
assuming 51 3 102 m s21 as the table value of v in
our calculations can introduce an additional
uncertainty in the error estimate in v of a little over
1%. Overall, based on these analyses, our
measurements seem to be giving satisfactory values
for v.
Table 1. List of tuning fork frequencies, t, L, l and calculated values of v.
✝
Column 2
Column 3
Column 4
Column 5
#
f1
(Hz)
t
(m)
L
(m)
l 5 6.173 L /t
(m)
v 5 lf1
21
(m s )
(corrected to a metre)
1
256
0.0072
0.151
19.55
5005
2
288
0.0072
0.140
16.80
4840
3
320
0.0073
0.134
15.18
4859
4
341.3
0.0072
0.128
14.05
4794
5
384
0.0072
0.123
12.97
4981
6
426.6
0.0073
0.117
11.58
4938
7
440
0.0073
0.113
10.80
4751
8
480
0.0071
0.109
10.33
4958
9
512
0.0073
0.106
9.50
4865
Average value of v 5 49.0 3 10 m s
2
2
Column 6
✝
Column 1
21
.
391
N E W A P P ROAC H E S
engineering details that go into making such a
simple piece of equipment as a tuning fork.
Received 28 November 1995, in final form 17 June 1996
References
[1] Kinsler L E, Frey A R, Coppens A B and Sanders
J V 1982 Fundamentals of Acoustics 3rd edn (New
York: John Wiley) pp 57–77
[2] Weidner R T and Sells R L 1973 Elementary
Classical Physics vol. 1 (Boston, MA: Allyn and
Bacon) pp 342–3
[3] Huntley H E 1967 Dimensional Analysis (New
York: Dover) pp 67–8
[4] Morse P M 1986 Vibration and Sound (New
York: American Institute of Physics for the Acoustical
Society of America) pp 156–63
Figure 2. Frequency against 1/l.
Figure 2 is a plot of f1 in Hz against 1/l in m21. As
expected, it is a straight line graph. The slope of this
line measures v. The extrapolated line passes
through the origin as expected.
Conclusions
This experiment can be adoped by institutions where
a collection of tuning forks made of the same metal
is available. It can be supplemented with a
discussion of the vibrations of a bar clamped at one
end. It gives an acceptable value of the speed of
sound in aluminium. It can be used to impress on
students all the scientific calculations and
392
[5] Stephens R W B and Bate A E 1966 Acoustics
and Vibrational Physics (London: Edward Arnold)
pp 79–81
[6] Rossing T D, Russel D A and Brown D E 1992
Acoustics of tuning forks Am. J. Phys. 60 620–6
[7] Morse P M and Ingard U K 1986 Acoustics
(Princeton, NJ: Princeton University Press) pp 181–5
[8] Stumpf F B 1980 Analytical Acoustics (Ann
Arbor, MI: Ann Arbor Science) pp 43–70
[9] Fletcher N H and Rossing T D 1991 The Physics
of Musical Instruments (New York: Springer) pp
33–64
[10] Hastings R B 1965 Laboratory Physics (St
Paul, MN: Bruce) pp 253–4
[11] Handbook of Chemistry and Physics
(Boca Raton, FL: CRC Press) pp B–7, D–187
1992