Gravitational Constant Determination by Using the Optical Method
In this experiment,we use the optical method of our measurements. It is an easy and convenient
way to make an accurate measurement in small scale system.Therefore,for this sensitive system,it
is an useful method to help us get the data.
I.
INTRODUCTION
In 1666,Isaac Newton found the gravitation by observing free fall and the movement of celestial bodies. The
relation is given by:
F =
GM m
d2
(1)
where G is the gravitational constant,and F and d are
the gravitation and the distance between two masses M
and m,respectively.
Hundred years later,English Scientist John Michell invented a torsion balance apparatus and tried to measure
the value of the gravitational constant.A few years later,
British Scientist Henry Cavendish is the first one that
yielded the accurate value of the gravitational constant
by using this apparatus. Since Cavendish’s significant
contribution of measuring the gravitational constant.
This experiment has been named after Cavendish. We
call this experiment ”Cavendish experiment”.
α ≡ [1 − (1 +
4r2 −3 ]
)2
d2
(3)
The small ball would started an underdamped oscillation by the quartz wire’s torsion and retarding force.Then
we put a flat mirror moving with wire. By the Principle of Optical Leverage, if displacement of mirror is θ ,the
reflection light would change with angle 2θ. In our experiment,when we move the balls from A1A2 to B1B2,total
displacement is 2θ and reflection light change with angle
4θ. We would obtain:
4θ =
∆x
L
(4)
where ∆x is the difference between two balance
positions,L is the distance between the mirror and the
projection ruler.
And the total torque of the wire is given by:
In the experiment we operated, we considered the net
torque of ball with mass m and radius r. It is attracted
by the ball with mass M and radius R.
τ = 2τ1 = κθ = I(
2π 2
2π
) θ = 2mr2 ( )2 θ
T
T
(5)
where κ is the torsion constant of wire, I is the
moment of inertia for the system, and T is the oscillation
period.
By arranging the above equation ,we can get the relation of G and the value we can measure.It is given by:
G=
π 2 rd2 ∆x
αM LT 2
(6)
In the experiment, we can find the value of ∆x , T,and
2
rd2
L. The value of παM
is calculated beforehand, it is
7.69 × 10−4 .That is , we can determine the value of G
by measuring ∆x , T,and L.
FIG. 1. Schematic diagram of the torque acting
The torque τ is given by the following equation:
τ = r1 × F1 + r2 × F2 = G
αM mr
d2
II.
METHOD
(2)
where τ is the torque, r and d is the distance shown
in figure, and α is the correction term of the small ball’s
attraction of two large balls ,it is defined as:
In the experiment, we use the following apparatus to
measure ∆x , T,and L. We have large and small a pair
of lead balls,shock absorber platform, laser light source,
and stopwatch(cellphone instead)
2
with the world standard value.
III.
RESULT
Here we use MaTLAB to fit the curve,we did it
twice,by using different distance.And we are going to
show our result and compare with the world standard
value.
We use the following form to fit the curve:
x = Ae−Bt sin(
2πt
+ D) + E
C
(7)
The result of first time:
FIG. 2. Schematic diagram of apparatus
The mass and radius of large balls are 1.5 kg and
0.032m ,the mass of small balls are 0.015 kg and 0.0069m
. r=0.05m, d=0.0465m.
(a) A=11.4,B=0.0564,c=9.934,D=1.563,E=53.1
FIG. 3. The position of balls (From top to bottom)
The only procedure we have to do is to measure the period of oscillation(T), difference of two balance point(∆x)
and the distance from apparatus to ruler(L). First,put
two large lead balls on A1 and A2 respectively.Wait for
a few minutes to reduce the effect of external force. Then
start to record the position of light spot per 30 seconds
for about 3 periods. After finishing recording,put two
large lead balls on B1 and B2 respectively and do the
same way again. Don’t forget to measure the distance
between apparatus to ruler to get the value of L. Finally
, using MATLAB to plot the position against time and
make fitline. We can get the information of ∆x and T.
By using equation(6), we can calculate the value of G.
We will operate twice to calculate the average and standard deviation of G. We will also compare G we derive
(b) A=-9.695,B=-0.05364,c=9.993,D=-5.384,E=67
FIG. 4. The position-time plot
From the data, T=9.964 min , ∆x=13.9 cm ,and
L=294.5cm. We get G= 1.016×10−10 N − m2 /kg 2
The result of second time:
3
IV.
DISCUSSION AND CONCLUSION
A.
(a) A=-22.37,B=-0.05931,c=9.59,D=45.3,E=59.54
(b) A=6.628,B=-0.05676,c=9.651,D=-0.5909,E=67.37
FIG. 5. The position-time plot
From the data, T=9.621 min , ∆x=7.83 cm ,and
L=196.5cm. We get G= 9.197×10−11 N − m2 /kg 2
Result and comparison:
Table.1 Comparison of world’s standard value
Average of G
9.679×10−11 (N − m2 /kg 2 )
Standard Derviation of G 6.809×10−12 (N − m2 /kg 2 )
World standard value of G 6.674×10−11 (N − m2 /kg 2 )
Discussion
In this experiment, we reproduced the Cavendish experiment by using optical method. However, we get the
larger value than we expected. The first reason of the
result might be the small value of the gravitational constant. Any small perturbation may cause huge effect.
The other reason is time scale, since small perturbation
would have large influence. It would took a lot of time to
become balance. We took about 3 periods of oscillation
at first case while we took about 4 periods of oscillation
at second case.The result shows that if we take for a long
time, we may get the more accurate value the gravitational constant.
This paragraph are going to discuss the precision of the
experiment. We recorded the data by keeping watching
the ruler . But we also have to check the stopwatch every
30 seconds for about 30 minutes to an hour. It would
some error while recording. We can take a film instead
and recording by replaying again. It may get the higher
precision of the experiment.
There are still one factor we want to discuss. It is
about us. The experiment operator’s mass would have
influence on our result or not. Suppose that there are a
90kg person,by gravitation law(Eq1). G,m is given, we
only have to compare the value of M/d2 .
Table.2 Different distance comparison
Large ball’s value:673.18
Distance(m)
The value of M/d2
0.2
2250
0.5
360
1
90
1.5
40
2
22.5
Though we are not a point of mass,we would not generate such a strong force.However, this table shows that if
we are too close to the apparatus,it would cause nonnegligible effect.Therefore,when recording the data,we
had better keep away from our apparatus to minimize
the error.
B.
Conclusion
Cavendish experiment use such a clever way to measure the gravitational constant. Since its convenience to
operate and calculate. It is still a universal way to for scientist to measure the gravitational constant.Though we
did the experiment roughly, the experiment inspired us
to design others experiments to measure small constant
or value.