PASCO AP-8215 User manual
!"#$$%"&'()*"+,-./0-1-, 23456
Instruction Manual and
Experiment Guide for the
PASCO scientific
Model AP-8215
6#7869%67:
##;$%
GRAVITATIONAL TORSION BALANCE
Attach to
Earth Ground.
GRAVITATIONAL
TORSION BALANCE
AP-8215
Gravitational Torsion Balance 012–06802B
The exclamation point within an equilateral
triangle is intended to alert the user of the
presence of important operating and
maintenance (servicing) instructions in the
literature accompanying the device.
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1
Introduction
The PASCO scientific AP-8215 Gravitational Torsion Balance
reprises one of the great experiments in the history of physics—
the measurement of the gravitational constant, as performed by
Henry Cavendish in 1798.
The Gravitational Torsion Balance consists of two 38.3 gram
masses suspended from a highly sensitive torsion ribbon and two
1.5 kilogram masses that can be positioned as required. The
Gravitational Torsion Balance is oriented so the force of gravity
between the small balls and the earth is negated (the pendulum is
nearly perfectly aligned vertically and horizontally). The large
masses are brought near the smaller masses, and the gravitational
force between the large and small masses is measured by
observing the twist of the torsion ribbon.
An optical lever, produced by a laser light source and a mirror
affixed to the torsion pendulum, is used to accurately measure the
small twist of the ribbon. Three methods of measurement are
possible: the final deflection method, the equilibrium method, and
the acceleration method.
A Little Background
The gravitational attraction of all objects toward the Earth is
obvious. The gravitational attraction of every object to every
other object, however, is anything but obvious. Despite the lack
of direct evidence for any such attraction between everyday
objects, Isaac Newton was able to deduce his law of universal
gravitation.
However, in Newton's time, every measurable example of this
gravitational force included the Earth as one of the masses. It was
therefore impossible to measure the constant, G, without first
knowing the mass of the Earth (or vice versa).
The answer to this problem came from Henry Cavendish in 1798,
when he performed experiments with a torsion balance,
measuring the gravitational attraction between relatively small
objects in the laboratory. The value he determined for Gallowed
the mass and density of the Earth to be determined. Cavendish's
experiment was so well constructed that it was a hundred years
before more accurate measurements were made.
Figure 1
Assembled Gravitational Torsion Balance,
ready to begin Henry Cavendish’s classic
experiment to determine the gravitational
constant
base with leveling feet
grounding
wire
large
masses
mirror on
pendulum bob
head of
torsion ribbon
zero adjust
knob
Newton’s law of universal
gravitation:
where m1and m2are the masses of
the objects, ris the distance between
them, and
G= 6.67 x 10-11 Nm2/kg2
F
=G
m
1
m
2
r
2
sight for
leveling
()*+,-*-,./*012.)3,./1'*0*/45 !"#6!%&!#'
2
Attach to
Earth Ground.
GRAVITATIONAL
TORSION BALANCE
AP-8215
Equipment
Included:
•GravitationalTorsionBalance
•supportbasewithlevelingfeet
•1.5kgleadballs(2)
•plasticplate
•replacementtorsionribbon
(part no. 004-06788)
•2-56x1/8Phillipsheadscrews(4)
•Phillipsscrewdriver(notshown)
Additional Required:
•laserlightsource(suchasthePASCOOS-9171He-NeLaser)
•meterstick
Figure 2
Equipment Included
1.5 kg lead
masses
replacement
torsion ribbon
plastic
demonstration
plate
aluminum
plate
leveling sight
large mass
swivel support
pendulum
mirror
zero adjust
knob
torsion ribbon
head
2-56x1/8
Phillips head
screws
optical grade
glass window
leveling feet
!"#6!%&!#' ()*+,-*-,./*012.)3,./1'*0*/45
3
Attach to
Earth Ground.
GRAVITATIONAL
TORSI ON BALANC E
AP-8215
Equipment Parameters
•Smallleadballs
Mass: 38.3 g + 0.2 g (m2)
Radius: 9.53 mm
Distance from ball center to torsion axis: d = 50 .0 mm
•Largeleadballs
Mass: 1500 g + 10 g (m1)
Radius: 31.9 mm
•Distancefromthecenterofmassofthelargeballtothe
center of mass of the small ball when the large ball is
against the aluminum plate and the small ball is in the
center position within the case: b= 46.5 mm (Tolerances
will vary depending on the accuracy of the horizontal
alignment of the pendulum.)
•Distancefromthesurfaceofthemirrortotheoutersurface
of the glass window: 11.4 mm
•TorsionRibbonMaterial:BerylliumCopper
Length: approx. 260 mm
Cross-section: .017 x .150 mm
Equipment Setup
Initial Setup
1. Place the support base on a flat, stable table that is located
such that the Gravitational Torsion Balance will be at least
5 meters away from a wall or screen.
Note: For best results, use a very sturdy table, such as an optics
table.
2. Carefully remove the Gravitational Torsion Balance from
the box, and secure it in the base.
3. Remove the front plate by removing the thumbscrews
(Figure 3), and carefully remove the packing foam from the
pendulum chamber.
Note: Save the packing foam, and reinstall it each time the
Gravitational Torsion Balance is transported.
4. Fasten the clear plastic plate to the case with the
thumbscrews.
IMPORTANT NOTES
➤The Gravitational Torsion Balance
is a delicate instrument. We
recommend that you set it up in a
relatively secure area where it is safe
from accidents and from those who
don’t fully appreciate delicate
instruments.
➤The first time you set up the
torsion balance, do so in a place
where you can leave it for at least one
day before attempting measurements,
allowing time for the slight elongation
of the torsion band that will occur
initially.
Keep the pendulum bob secured in
the locking mechanisms at all times,
except while setting up and
conductingexperiments.
Do not touch the mirror on the pendulum.
Figure 3
Removing a plate from the chamber box
pendulum
chamber
aluminum plate
pendulum
bob
()*+,-*-,./*012.)3,./1'*0*/45 !"#6!%&!#'
4
Vertical Adjustment of the Pendulum
The base of the pendulum should be flush with the floor of the
pendulum chamber. If it is not, adjust the height of the pendulum:
1. Grasp the torsion ribbon head and
loosen the Phillips retaining screw
(Figure 6a).
2. Adjust the height of the pendulum
by moving the torsion ribbon head
up or down so the base of the
pendulum is flush with the floor of
the pendulum chamber
(Figure 6b).
3. Tighten the retaining (Phillips
head) screw.
Leveling the Gravitational Torsion Balance
1. Release the pendulum from the locking mechanism by
unscrewing the locking screws on the case, lowering the
locking mechanisms to their lowest positions (Figure 4).
Figure 5
Using the leveling sight to level the
Gravitational Torsion Balance.
Look through the
sight to view the
reflection of the
pendulum bob in
the mirror.
Pendulum bob
must be
centered over
the mirror.
pendulum
mirror
torsion ribbon
torsion ribbon
head
Figure 4
Lowering the locking mechanism to release the pendulum bob arms
Turn locking
screws clockwise.
locking
mechanisms
pendulum
bob arm
2. Adjust the feet of the base until the pendulum is centered in
the leveling sight (Figure 5). (The base of the pendulum will
appear as a dark circle surrounded by a ring of light).
3. Orient the Gravitational Torsion Balance so the mirror on
the pendulum bob faces a screen or wall that is at least 5
meters away.
!"SIDE,
CUTAWAY
VIEW
"
!
Figure 6
Adjusting the height of the pendulum bob
The bottom of the pendulum
bob should be flush with the
floor of the chamber.
Grasp the torsion ribbon head
and loosen the Phillips screw.
!"#6!%&!#' ()*+,-*-,./*012.)3,./1'*0*/45
5
Rotational Alignment of the Pendulum Bob Arms
(Zeroing)
The pendulum bob arms must be centered rotationally in the case
— that is, equidistant from each side of the case (Figure 7). To
adjust them:
1. Mount a metric scale on the wall or other projection surface
that is at least 5 meters away from the mirror of the
pendulum.
2. Replace the plastic cover with the aluminum cover.
3. Set up the laser so it will reflect from the mirror to the
projection surface where you will take your measurements
(approximately 5 meters from the mirror). You will need to
point the laser so that it is tilted upward toward the mirror
and so the reflected beam projects onto the projection surface
(Figure 8). There will also be a fainter beam projected off the
surface of the glass window.
Note: Vertical adjustment is only necessary at initial setup and
when you change the torsion ribbon or if someone has loosened
the retaining screw by mistake; it is not normally done during
each experimental setup.
L
Figure 8
Setting up the optical lever
SIDE VIEW
TOP VIEW
L
laser
reflected beam
(from mirror)
screen with scale
Figure 7
Aligning the pendulum bob rotationally
TOP, CUTAWAY VIEW
small mass
case plates
The pendulum bob arm
must be centered
rotationally between the
plates.
()*+,-*-,./*012.)3,./1'*0*/45 !"#6!%&!#'
6
3. Rotationally align the case by rotating it until the laser beam
projected from the glass window is centered on the metric
scale (Figure 9).
4. Rotationally align the pendulum arm:
a. Raise the locking mechanisms by turning the locking
screws until both of the locking mechanisms barely
touch the pendulum arm. Maintain this position for a
few moments until the oscillating energy of the
pendulum is dampened.
b. Carefully lower the locking mechanisms slightly so the
pendulum can swing freely. If necessary, repeat the
dampening exercise to calm any wild oscillations of the
pendulum bob.
c. Observe the laser beam reflected from the mirror. In the
optimally aligned system, the equilibrium point of the
oscillations of the beam reflected from the mirror will be
vertically aligned below the beam reflected from the
glass surface of the case (Figure 9).
d. If the spots on the projection surface (the laser beam
reflections) are not aligned vertically, loosen the zero
adjust thumbscrew, turn the zero adjust knob slightly to
refine the rotational alignment of the pendulum bob
arms (Figure 10), and wait until the movement of the
pendulum stops or nearly stops.
e. Repeat steps 4a – 4c as necessary until the spots are
aligned vertically on the projection surface.
5. When the rotational alignment is complete, carefully tighten
the zero adjust thumbscrew, being careful to avoid jarring the
system.
Figure 9
Ideal rotational alignment (zeroing) of the
pendulum
location of the
projected laser
beam from the glass
window
zero adjust thumbscrew
zero adjust knob
Figure 10
Refining the rotational alignment of the
pendulum bob
Hints for speedier rotational alignments:
•Dampenanywildoscillationsofthependulumbobwiththe
locking mechanisms, as described;
•Adjusttherotationalalignmentofthependulumbobusing
small, smooth adjustments of the zero adjust knob;
•Exercisepatienceandfinesseinyourmovements.
location of the
projected laser beam
from the mirror
!"#6!%&!#' ()*+,-*-,./*012.)3,./1'*0*/45
7
Attach to
Earth Ground.
GRAV
IT
TORSION
B
Setting up for the Experiment
1. Take an accurate measurement of the distance from the
mirror to the zero point on the scale on the projection
surface (L) (Figure 8). (The distance from the mirror surface
to the outside of the glass window is 11.4 mm.)
Note: Avoid jarring the apparatus during this setup procedure.
2. Attach copper wire to the grounding screw (Figure 11), and
ground it to the earth.
3. Place the large lead masses on the support arm, and rotate
the arm to Position I (Figure 12), taking care to avoid
bumping the case with the masses.
4. Allow the pendulum to come to resting equilibrium.
5. You are now ready to make a measurement using one of
three methods: the final deflection method, the equilibrium
method, or the acceleration method.
Note: The pendulum may require several hours to reach resting
equilibrium. To shorten the time required, dampen the oscillation
of the pendulum by smoothly raising the locking mechanisms up
(by turning the locking screws) until they just touch the crossbar,
holding for several seconds until the oscillations are dampened,
and then carefully lowering the locking mechanisms slightly.
Figure 11
Attaching the grounding strap to the
grounding screw
copper wire to
earth ground
grounding
screw
Figure 12
Large Masses:
Position I
Large Masses:
Position II
Mirror
Light beam
Measuring theGravitational
Constant
Overview of the Experiment
The gravitational attraction between a 15 gram mass and a 1.5 kg
mass when their centers are separated by a distance of
approximately 46.5 mm (a situation similar to that of the
Gravitational Torsion Balance ) is about 7 x 10 -10 newtons. If this
doesn’t seem like a small quantity to measure, consider that the
weight of the small mass is more than two hundred million times
this amount.
case
glass
window
small
mass
Moving the large masses into Position I
Gravitational Torsion Balance 012–06802B
8
d
b
The enormous strength of the Earth's attraction for the small
masses, in comparison with their attraction for the large masses, is
what originally made the measurement of the gravitational
constant such a difficult task. The torsion balance (invented by
Charles Coulomb) provides a means of negating the otherwise
overwhelming effects of the Earth's attraction in this experiment.
It also provides a force delicate enough to counterbalance the tiny
gravitational force that exists between the large and small masses.
This force is provided by twisting a very thin beryllium copper
ribbon.
The large masses are first arranged in Position I, as shown in
Figure 12, and the balance is allowed to come to equilibrium. The
swivel support that holds the large masses is then rotated, so the
large masses are moved to Position II, forcing the system into
disequilibrium. The resulting oscillatory rotation of the system is
then observed by watching the movement of the light spot on the
scale, as the light beam is deflected by the mirror.
Any of three methods can be used to determine the gravitational
constant, G, from the motion of the small masses. In Method I,
the final deflection method, the motion is allowed to come to
resting equilibrium—a process that requires several hours — and
the result is accurate to within approximately 5%. In method II,
the equilibrium method, the experiment takes 90 minutes or more
and produces an accuracy of approximately 5% when graphical
analysis is used in the procedure. In Method III, the acceleration
method, the motion is observed for only 5 minutes, and the result
is accurate to within approximately 15%.
METHODI:Measurement by FinalDeflection
Setup Time: ~ 45 minutes; Experiment Time: several hours
Accuracy: ~ 5%
Theory
With the large masses in Position I (Figure 13), the gravitational
attraction, F, between each small mass (m2) and its neighboring
large mass (m1) is given by the law of universal gravitation:
F = Gm1m2/b2 (1.1)
where b= the distance between the centers of the two
masses.
Large Masses:
Position I
Large Masses:
Position II
Figure 13
Origin of variables
b
and
d
➤Note: 5% accuracy is possible in
Method I if the experiment is set up on a
sturdy table in an isolated location
where it will not be disturbed by
vibration or air movement.
➤Note: 5% accuracy is possible in
Method II if the resting equilibrium
points are determined using a graphical
analysis program.
012–06802B Gravitational Torsion Balance
9
The gravitational attraction between the two small masses and
their neighboring large masses produces a net torque ( grav) on
the system:
grav = 2Fd (1.2)
where d is the length of the lever arm of the
pendulum bob crosspiece.
Since the system is in equilibrium, the twisted torsion band
must be supplying an equal and opposite torque. This torque
(band) is equal to the torsion constant for the band (κ) times the
angle through which it is twisted (
θ
), or:
band = –
κθ
.(1.3)
Combining equations 1.1, 1.2, and 1.3, and taking into account
that grav = – band, gives:
κθ
= 2dGm1m2/b2
Rearranging this equation gives an expression for G:
(1.4)
To determine the values of
θ
and κ— the only unknowns in
equation 1.4 — it is necessary to observe the oscillations of the
small mass system when the equilibrium is disturbed. To
disturb the equilibrium (from S1), the swivel support is rotated
so the large masses are moved to Position II. The system will
then oscillate until it finally slows down and comes to rest at a
new equilibrium position (S2) (Figure 14).
At the new equilibrium position S2, the torsion wire will still be
twisted through an angle
θ
, but in the opposite direction of its
twist in Position I, so the total change in angle is equal to 2
θ
.
Taking into account that the angle is also doubled upon
reflection from the mirror (Figure 15):
Δ
S = S2– S1,
4
θ
=
Δ
S/L or
θ
=
Δ
S/4L(1.5)
T
20
60
S
(cm)
Time
(min)
s
2
s
1
Figure 14
Graph of Small Mass Oscillations
G=κθb2
2dm1m2
S1S2
ΔS
2
θ
L
Figure 15
Diagram of the experiment showing the
optical lever.
2θ»tan(2θ)=ΔS
2L
Position II
Position I
Gravitational Torsion Balance 012–06802B
10
The torsion constant can be determined by observing the period
(T) of the oscillations, and then using the equation:
T2= 4
π
2I/
κ
(1.6)
where I is the moment of inertia of the small mass system.
The moment of inertia for the mirror and support system for the
small masses is negligibly small compared to that of the
masses themselves, so the total inertia can be expressed as:
(1.7)
Therefore:
(1.8)
Substituting equations 1.5 and 1.8 into equation 1.4 gives:
(1.9)
All the variables on the right side of equation 1.9 are known or
measurable:
r=9.55 mm
d= 50 mm
b= 46.5 mm
m1= 1.5 kg
L= (Measure as in step 1 of the setup.)
By measuring the total deflection of the light spot (
Δ
S) and the
period of oscillation (T), the value of Gcan therefore be
determined.
Procedure
1. Once the steps for leveling, aligning, and setup have been
completed (with the large masses in Position I), allow the
pendulum to stop oscillating.
2. Turn on the laser and observe the Position I end point of
the balance for several minutes to be sure the system is at
equilibrium. Record the Position I end point (S1) as
accurately as possible, and indicate any variation over
time as part of your margin of error in the measurement.
G=π2ΔSb2(d2+25
25r2)
T2m1Ld
I=2m2(d2+25
25r2)
κ=8π2m2
d2+25
25r2
T2
012–06802B Gravitational Torsion Balance
11
3. Carefully rotate the swivel support so that the large
masses are moved to Position II. The spheres should be
just touching the case, but take care to avoid knocking the
case and disturbing the system.
Note: You can reduce the amount of time the pendulum
requires to move to equilibrium by moving the large
masses in a two-step process: first move the large masses
and support to an intermediate position that is in the
midpoint of the total arc (Figure 16), and wait until the
light beam has moved as far as it will go in the period;
then move the sphere across the second half of the arc
until the large mass support just touches the case. Use a
slow, smooth motion, and avoid hitting the case when
moving the mass support.
4. Immediately after rotating the swivel support, observe the
light spot and record its position (S1).
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 Tis squared in the calculation of G.
6. Wait until the oscillations stop, and record the resting
equilibrium point (S2).
Analysis
1. Use your results and equation 1.9 to determine the value
of G.
2. The value calculated in step 2 is subject to the following
systematic 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
S
1
S
2
L
Figure 16
Two-step process of moving the large
masses to reduce the time required to stop
oscillating
Position I Position II
intermediate
position
Gravitational Torsion Balance 012–06802B
12
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, F0is given by the gravitational law, which
translates, 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, b, that is
equal to the ratio of the magnitude of ƒ to that of F. Using
the equation F = Gm1m2/b2, it can be determined that:
b = b3/(b2+ 4d2)3/2
From Figure 17,
Fnet = F - f = F - bF = F(1 - b)
where Fnet 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 = G0(1- b)
where Gis your experimentally determined value for the
gravitational constant, and G0is corrected to account for
the systematic error.
Finally,
G0= G/(1- b)
Use this equation with equation 1.9 to adjust your
measured value.
F0=Gm2m1
(b2+4d2)
f=Gm2m1b
(b2+4d2)(b2+4d2)
1
2
=βF
Φ
d
b
F
0
F
f
Figure 17
Correcting the measured value of
G
sinΦ=b
(b2+4d2)12
12
f=F0sinΦ
012–06802B Gravitational Torsion Balance
13
METHODII:Measurement by Equilibrium
Positions
Observation Time: ~ 90+ minutes
Accuracy: ~ 5 %
Theory
When the large masses are placed on the swivel support and
moved to either Position I or Position II, the torsion balance
oscillates for a time before coming to rest at a new equilibrium
position. This oscillation can be described by a damped sine
wave with an offset, where the value of the offset represents
the equilibrium point for the balance. By finding the
equilibrium point for both Position I and Position II and taking
the difference, the value of
Δ
Scan be obtained. The remainder
of the theory is identical to that described in
Method I.
Procedure
1. Set up the experiment following steps 1–3 of Method I.
2. Immediately after rotating the swivel support to Position
II, observe the light spot. Record the position of the light
spot (S) and the time (t) every 15 seconds. Continue
recording the position and time for about 45 minutes.
3. Rotate the swivel support to Position I. Repeat the
procedure described in step 2.
Note: Although it is not imperative that step 3 be performed
immediately after step 2, it is a good idea to proceed with
it as soon as possible in order to minimize the risk that the
system will be disturbed between the two measurements.
Waiting more than a day to perform step 3 is not advised.
Analysis
1. Construct a graph of light spot position versus time for
both Position I and Position II. You will now have a
graph similar to Figure 18.
2. Find the equilibrium point for each configuration by
analyzing the corresponding graphs using graphical
analysis to extrapolate the resting equilibrium points S1
and S2(the equilibrium point will be the center line about
which the oscillation occurs). Find the difference between
60
S
(cm)
Time
(min)
S
2
S
1
0
20
Figure 18
Typical pendulum oscillation pattern
showing equilibrium positions
Note: To obtain an accuracy of 5%
with this method, it is important
to use graphical analysis of the
position and time data to
extrapolate the resting equilibrium
positions, S1and S2.
Gravitational Torsion Balance 012–06802B
14
the two equilibrium positions and record the result as
Δ
S.
3. Determine the period of the oscillations of the small mass
system by analyzing the two graphs. Each graph will
produce a slightly different result. Average these results
and record the answer as T.
4. Use your results and equation 1.9 to determine the value
of G.
5. The value calculated in step 4 is subject to the same
systematic error as described in Method I. Perform the
correction procedure described in that section (Analysis,
step 3) to find the value of G0.
METHODIII:
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 (m2) and its neighboring large
mass (m1) is given by the law of universal gravitation:
F = Gm1m2/b2(3.1)
This force is balanced by a torque from the twisted torsion
ribbon, so that the system is in equilibrium. The angle of twist,
θ
, 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 significantly
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
012–06802B Gravitational Torsion Balance
15
changed, the total force (Ftotal) that is now acting to accelerate
the small masses is equal to twice the original gravitational
force from the large masses, or:
Ftotal = 2F = 2Gm1m2/b2(3.2)
Each small mass is therefore accelerated toward its
neighboring large mass, with an initial acceleration (a0) that is
expressed in the equation:
m2a0= 2Gm1 m2/b2(3.3)
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:
G = b2a0/2m1(3.4)
The acceleration is measured by observing the displacement of
the light spot on the screen. If, as is shown in Figure 19:
Δ
s=thelineardisplacementofthesmallmasses,
d=thedistancefromthecenterofmassofthesmall
masses to the axis of rotation of the torsion
balance,
Δ
S=thedisplacementofthelightspotonthescreen,
and
L=thedistanceofthescalefromthemirrorofthe
balance,
then, taking into account the doubling of the angle on
reflection,
Δ
S =
Δ
s(2L/d ) (3.5)
Using the equation of motion for an object with a constant
S
1
S
2
ΔS
2
2θ
L
Figure 19
Source of data for calculations in
Method III
()*+,-*-,./*012.)3,./1'*0*/45 !"#6!%&!#'
16
acceleration (x = 1/2 at2), the acceleration can be calculated:
a0= 2
Δ
s/t2=
Δ
Sd/t2L(3.6)
By monitoring the motion of the light spot over time, the
acceleration can be determined using equation 3.6, and the
gravitational constant can then be determined using equation
3.4.
Procedure
1. Begin the experiment by completing steps 1–3 of the
procedure detailed in Method I.
2. Immediately after rotating the swivel support, observe the
light spot. Record the position of the light spot (S) and the
time (t) every 15 seconds for about two minutes.
Analysis
1. Construct a graph of light spot displacement
(
Δ
S = S - S1) versus time squared (t2), with t 2on the
horizontal axis (Figure 20). Draw a best-fit line through
the observed data points over the first minute of
observation.
2. Determine the slope of your best-fit line.
3. Use equations 3.4 and 3.6 to determine the gravitational
constant.
4. The value calculated in step 3 is subject to a systematic
error. The small sphere is attracted not only to its
neighboring large sphere, but also to the more distant
large sphere, although with a much smaller force. Use the
procedure detailed in Method I (Analysis, step 3) to
correct for this force.
ΔS
t
2
(sec
2
)
0
1
2
3
4
5
6
7
8
9
225 2025
900 3600 5625 8100 11025
BEST FIT LINE
CURVE
THROUGH
DATA
Figure 20
Sample data and best-fit line
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