PASCO AP-8215A Gravitational Torsion Balance - Manual (Page 16)

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G r a v i t a t i o n a l T o r s i o n B a l a n c e M E T H O D I I I : M e a s u r e m e n t b y A c c e l e r a t i o n
12
METHOD III: Measurement by Acceleration
Observation Time: ~ 5 minutes
Accuracy: ~ 15%
Theory
With the large masses in Position I, the gravitational attraction, F, between each small mass (m
2
) and its neigh-
boring large mass (m
1
) is given by the law of universal gravitation:
This force is balanced by a torque from the twisted torsion ribbon, so that the system is in equilibrium. The angle
of twist,
, is measured by noting the position of the light spot where the reflected beam strikes the scale. This
position is carefully noted, and then the large masses are moved to Position II. The position change of the large
masses disturbs the equilibrium of the system, which will now oscillate until friction slows it down to a new
equilibrium position.
Since the period of oscillation of the small masses is long (approximately 10 minutes), they do not move signif-
cantly when the large masses are first moved from Position I to Position II. Because of the symmetry of the
setup, the large masses exert the same gravitational force on the small masses as they did in Position I, but now in
the opposite direction. Since the equilibrating force from the torsion band has not changed, the total force (F
total
)
that is now acting to accelerate the small masses is equal to twice the original gravitational force from the large
masses, or:
Each small mass is therefore accelerated toward its neighboring large mass, with an initial acceleration (a
0
) that
is expressed in the equation:
Of course, as the small masses begin to move, the torsion ribbon becomes more and more relaxed so that the
force decreases and their acceleration is reduced. If the system is observed over a relatively long period of time,
as in Method I, it will be seen to oscillate. If, however, the acceleration of the small masses can be measured
before the torque from the torsion ribbon changes appreciably, equation 3.3 can be used to determine G. Given
the nature of the motion—damped harmonic—the initial acceleration is constant to within about 5% in the first
one tenth of an oscillation. Reasonably good results can therefore be obtained if the acceleration is measured in
the first minute after rearranging the large masses, and the following relationship is used:
F G
m
1
m
2
b
2
-------------- 3.1   =
F
total
2F 2G
m
1
m
2
b
2
-------------- 3.2   = =
m
2
a
0
2G
m
1
m
2
b
2
-------------- 3.3   =
G b
2
a
0
2m
1
---------- 3.4   =