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Fig. 1 Oscillations of a reversible pendulum around the
suspen-
sion points H1a nd H2.
MechanicsOscillations
Simple and compound pendulum
LD
Physics
Leaflets
Determining
the gravitational acceleration
w ith a revers ib le pend ulum
Objects of the experiments
Measuring the oscillation periods T1and T2of a reversible
pendulum for two suspe nsion points.
Tuning the reversible pe ndulum to the sa me o sc illation
period .
Determining the g ravitational a cc eleration from the osc
illation pe riod and the reduced length of pend ulum.
Principles
Compound pendulum:
If a compound pendulum oscillates around its rest position
with s ma ll deflections , the eq uation of motion is:
J..
+msg=0 (I)J: mo ment o f inertia around the axis o f os
cillation,
s: distance between the axis of oscillation and the centre
of
mass , g: g ravitational ac celeration, m: mass of the
pendulum
The reduced length o f the co mpound pendulum is defined a s
the quantity
sr=J
ms(II)
bec aus e its os cillation period
T=2srg (III)
co rrespo nds to tha t of a simple pend ulum with the length
sr.
The moment o f inertia J of the co mpound pend ulum is, a cc
ord-
ing to the parallel axis theorem,
J=JS +ms2 (IV).JS : moment of inertia a round the c entre of ma
ss axis
Therefore the reduc ed leng th of p end ulum is
sr=JS
ms+s (V).
Reversible pendulum:
The reversible pend ulum is a pa rticular type o f the co mpo
und
pendulum. There a re tw o e dg es H1and H2that allow to choos
e
the susp ension point. Two mas ses m1= 1000 g and
m2= 1400 g on the s traight line H
1H
2 can be shifted so that
the o sc illation period is tuna ble. The go a l of the tuning
is to
achieve equal oscillation periods around both edges. In this
ca se, the reduced length of pendulum is eq ual to the distanc
e
d= 99.4 cm b etwe en the edge s. This latter statement c an
be
understood from the follow ing c onsideration:
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Be ca use o f Eq. (II) the pendulum osc illate s a round b oth
ed ges
H1and H2a t the sa me os cillation period T1= T2if the reduce
d
lengths of pe ndulum sr,1and sr,2a gree. In this c as e w e
have
JS
ms1+s1=
JS
ms2+s2 (VI).
The relation betw een the d ista nces s1und s2between the
two
edges and the c entre of mass is g iven by
s1+s2=d (VII)
beca use the centre of mas s S is located o n the connecting
line H1H2for rea so ns of s ymmetry.
From Eqs. (VI) and (VII) a quadratic equation is obtained
for
determining s1. Its s olution is
s1=d
2d
2
4
JS
m(VIII).
If s1 is inserted in Eq. (V) instead of s, the reduced length
of
pendulum of the tuned reversible pendulum is obtained:
sr= d (IX).
Determining the gravitational acceleration:
For the tuned reversible pend ulum
T2=4 2d
g(X)
follow s from Eq s. (III) a nd (IX). The d ista nc e d between
the
ed ge s is know n to a h igh prec ision. After the osc illa tion
period
T of the tuned pendulum has been determined, the gravi-
tational ac celeration gis o bta ined from Eq . (X).
Setup
The e xperimenta l setup is illustra ted in Fig. 2.
If neces sa ry, a ss emble the reversible pend ulum ac co
rdingto the instruction sheet, and mount the wall holder on a
sta ble, non-vibrating w all.
Make marks on the pendulum rod at x1 = 25 cm andx2= 10 cm, 15
cm, 20 cm etc. (sta rting from the edg e H1,
take into account the radius r= 5 cm of the masses, see
Fig. 3)
Fix the mass m1at the po sition x1= 25 cm.
Carrying out the experiment
Ca refully ha ng the reversible pend ulum on the ed ge bea r-ing
o f the w all holder with the edg e H1a nd fix the mass m2at the
distanc e x2= 50 cm.
Ca utious ly de flec t the low er end of the reversible
pendulumpa rallel to the wa ll, a nd ma ke it o sc illate , if poss
ible, without
any vibrations.
Measure the length of 50 oscillations and take it down asmea
sured value 50 T1.
Apparatus
1 reversible pendulum . . . . . . . . . . . 346 111
1 steel tape mea sure, 2m . . . . . . . . . . 311 77
1 sto pc loc k I, 30s /15 min . . . . . . . . . . 313 07
Fig. 2 Experimental setup for determining the gravitational ac
cel-
eration w ith a reversible p endulum.
Fig . 3 D is t a nc e ma rks x1and x2on the p endulum rod .
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Hang the reversible pend ulum on the ed ge bea ring w ith
theedge H2, and measure the period 50 T2.
S lide the mass m2to the p os ition x2= 55 cm, and meas ure50
T2a t first a nd then 50 T1.
Slide the mass m2towards the edge H2in steps of 5 cm;ea ch time
mea sure the two o sc illa tion period s. P lot T1
2and
T22
a s functions o f x2a nd, if necess ary, repea t the meas
-urement of the os cilla tion p eriod s.
Next slide the mass m2 towards the edge H1 in steps of5 cm s
tarting from x2= 45 cm, a nd meas ure the two os cil-
la tion period s e a ch time.
Measuring example
Evaluation and results
The two m ea sured curves T12(x2) and T2
2(x2) intersect near
x2= 30 cm and x2= 65 cm (se e Fig. 4). The enlarged se ctionsin
Figs . 5 and 6 sho wn tha t the curves interse ct a t T2= 4.039
s2
and T2= 4.014 s2, respectively. With the mean value
T2= 4.027 s2Eq . (X) g ives
g=4 2d
T2=9.74
m
s 2
Ta ble 1: Os cillation periods T1 and T2 around the edges H1and
H2, respectively, as functions of the distance x2between
the mass m2and the edge H1.
x2cm
50T1
sT1
2
s 250
T2
sT2
2
s 2
20 106.0 4.494 101.9 4.153
25 103.1 4.252 101.2 4.097
30 100.8 4.064 100.6 4.048
35 99.4 3.952 100.1 4.008
40 98.8 3.905 99.8 3.984
45 98.2 3.857 99.6 3.968
50 97.9 3.834 99.5 3.960
55 98.3 3.865 99.6 3.968
60 99.1 3.928 99.8 3.984
65 99.9 3.992 100.0 4.000
70 101.1 4.088 100.7 4.056
75 102.2 4.178 101.7 4.137
80 103.2 4.260 102.2 4.178
85 104.8 4.393 103.6 4.293
Fig . 4 Sq uared osci lla t ion periods a round the edges H1() a
nd H2( )as functions of the distanc e x2between
the mass m2and the edge H1.
F ig . 5 Enlarged sect ion of Fig . 4 around x2= 30 cm w ith a
non-
linea r interpola tion o f the me as ured values. P oint of
inter-
section: (x2= 31 cm, T2= 4.039 s2).
Fig . 6 Enlarged sect ion of Fig . 4 around x2= 65 cm w ith a
non-
linea r interpola tion o f the me as ured values. P oint of
inter-
section: (x2= 66 cm, T2= 4.014 s2).
LD DIDACTIC G mb H Leyboldstrasse 1 D-50354 Hrth P hon e (02233)
604-0 Telefa x (02233) 604-222 E-mail: info@ld-dida ctic.d e by LD
DIDAC TIC G mbH P rinted in the Federa l Repub lic o f G erma
ny
Technica l alterations reserved
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