®
Gravitational Torsion Balance
METHOD I: Measurement by Final Deflection
10
012-11032C
5.
Use a stop watch to determine the time required for one period of oscillation (
T
). For greater accuracy,
include several periods, and then find the average time required for one period of oscillation.
Note:
The accuracy of this period value (
T
) is very important, since the
T
is squared in the calculation of G.
6.
Wait until the oscillations stop, and record the resting equilibrium point (S
2
).
Analysis
1.
Use your results and equation 1.9 to determine the value of
G
.
The value calculated in step 2 is subject to the following sys-
tematic error. The small sphere is attracted not only to its
neighboring large sphere, but also to the more distant large
sphere, though with a much smaller force. The geometry for
this second force is shown in Figure 17 (the vector arrows
shown are not proportional to the actual forces).
From Figure 17,
The force, F
0
is given by the gravitational law, which trans-
lates, in this case, to:
and has a component ƒ that is opposite to the direction of the force
F
:
This equation defines a dimensionless parameter,
, that is equal to the ratio of the magnitude of ƒ to that of
F
.
Using the equation
F
= Gm
1
m
2
/b
2
, it can be determined that:
From Figure 17,
F
net
= F - f = F -
F = F(1 -
)
where
F
net
is the value of the force acting on each small sphere from
both
large masses, and
F
is the force of
attraction to the nearest large mass only.
Similarly,
G = G
0
(
1
-
)
where
G
is your experimentally determined value for the gravitational constant, and
G
0
is corrected to account
for the systematic error.
Finally,
G
0
= G/(
1
-
)
Use this equation with equation 1.9 to adjust your measured value.
Φ
d
b
F
0
F
f
Figure 17: Correcting the
measured value of G
f
F
0
sin
=
F
0
Gm
2
m
1
b
2
4
d
2
+
-------------------------
=
f
Gm
2
m
1
b
b
2
4
d
2
+
b
2
4
d
2
+
1
2
---
------------------------------------------------------
F
=
=
b
3
b
2
4
d
2
+
3
2
---
----------------------------
=