WPI Physics Dept. Intermediate Lab 2651 Cavendish...

WPI Physics Dept.

Intermediate Lab 2651

Cavendish experiment: Determination of the

universal gravitational constant

March 11, 2017

Index:

1. Introduction, p. 2

2. Pre-lab Exercises, p. 3

3. Instrument Manual, p. 5

1

1 Introduction

There are four fundamental interactions in nature: strong, electromagnetic,weak, and gravitational forces. The gravitational interaction is consideredthe weakest of all, and relatively speaking is 10−36 times smaller than theelectromagnetic force. Massive bodies experience a gravitational attractiveforce, which involves their separation and mass. For two point masses,m1, m2, which are separated by a distance vector ~R, the magnitude of thegravitational force is

F = Gm1m2

R2(1)

In Eq. 1, G is called the Universal Gravitational constant. Its numericalvalue was determined by Henry Cavendish in 1798 by using a torsion pendu-lum. The details of the experiment can be found in the PASCO instructionmanual attached with this document. Please note that:

1. The instrument is very delicate. Please do not touch any parts of theinstruments other than the swivel support for the large spheres. Whileswitching the large spheres from POSITION-I to POSITION-II, pleasedo very carefully so that the large spheres gently touch the aluminiumwall, and do not bump on the wall.

2. Please read the manual carefully and work on the pre-lab questionsBEFORE you come to the lab.

2

2 Pre-lab Exercises

1. Derive an expression for the Universal Gravitational constant, G inthefollowing form:

G = K∆S

LT 2

where L is the distance between the mirror and the screen, ∆S isthe distance between the laser spots when the pendulum is set intoopposite equilibrium positions, T is the period of oscillation of thependulum, and K is a constant for the apparatus as

K =π2b2

Md(d2 +

2

5r2) = 7.2173 × 10−4m3/kg

with b = 46.5 mm is the distance between a large and a small spherewhen they are in their closest separation, M = 1.5 kg is the mass ofthe big sphere, r = 9.55 mm is the radius of the small sphere, d = 50mm is the lever arm of the torsion pendulum from the center.

2. Starting from Newton’s Second Law of Motion, derive an expressionfor the time period of a torsion pendulum under small angle approxi-mation:

T = 2π

√I

κ

where I is the moment of inertia of the torsion pendulum and κ is thetorsion constant of the metal ribbon holding the pendulum.

3. How does the period of the torsion pendulum in Prelab question 2change if a velocity dependent damping force, f = −βθ̇ is introducedin the system?

4. Drawing a suitable ray diagram, prove the following : if the directionof the incident beam to a mirror is kept constant, the reflected beamfrom the mirror rotates through twice the angle of rotation of themirror itself.

5. In the Cavendish Experiment, when the two large lead spheres arebrought closest to the small lead spheres, the small spheres experiencea weaker force from the furthest big spheres in addition to a strongerforce from the closest spheres. Draw a suitable force diagram of thearrangement to show the relevant forces on a small sphere, and hencederive a corrected expression for G.

3

6. Derive an expression for the intensity of gravitational field at a pointdue to a sphere by using the Gausss Law of gravity. Hence show thatthe two lead spheres in Cavendish Experiment can be considered aspoint masses in calculating the graviational force between them.

4

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Instruction Manual andExperiment Guide for thePASCO scientificModel AP-8215

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GRAVITATIONAL TORSION BALANCE

Attach to

Earth Ground.

GRAVITATIONAL

TORSION BALANCEAP-8215

Gravitational Torsion Balance 012–06802B

The exclamation point within an equilateraltriangle is intended to alert the user of thepresence of important operating andmaintenance (servicing) instructions in theliterature accompanying the device.

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1

IntroductionThe PASCO scientific AP-8215 Gravitational Torsion Balancereprises one of the great experiments in the history of physics—the measurement of the gravitational constant, as performed byHenry Cavendish in 1798.

The Gravitational Torsion Balance consists of two 38.3 grammasses suspended from a highly sensitive torsion ribbon and two1.5 kilogram masses that can be positioned as required. TheGravitational Torsion Balance is oriented so the force of gravitybetween the small balls and the earth is negated (the pendulum isnearly perfectly aligned vertically and horizontally). The largemasses are brought near the smaller masses, and the gravitationalforce between the large and small masses is measured byobserving the twist of the torsion ribbon.

An optical lever, produced by a laser light source and a mirroraffixed to the torsion pendulum, is used to accurately measure thesmall twist of the ribbon. Three methods of measurement arepossible: the final deflection method, the equilibrium method, andthe acceleration method.

A Little BackgroundThe gravitational attraction of all objects toward the Earth isobvious. The gravitational attraction of every object to everyother object, however, is anything but obvious. Despite the lackof direct evidence for any such attraction between everydayobjects, Isaac Newton was able to deduce his law of universalgravitation.

However, in Newton's time, every measurable example of thisgravitational force included the Earth as one of the masses. It wastherefore impossible to measure the constant, G, without firstknowing 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 smallobjects in the laboratory. The value he determined for G allowedthe mass and density of the Earth to be determined. Cavendish'sexperiment was so well constructed that it was a hundred yearsbefore more accurate measurements were made.

Figure 1Assembled Gravitational Torsion Balance,ready to begin Henry Cavendish’s classicexperiment to determine the gravitationalconstant

base with leveling feet

groundingwire

largemasses

mirror onpendulum bob

head oftorsion ribbon

zero adjustknob

Newton’s law of universalgravitation:

where m1 and m2 are the masses ofthe objects, r is the distance betweenthem, andG = 6.67 x 10-11 Nm2/kg2

F = G m1m2r2

sight forleveling

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2

Attach to

Earth Ground.

GRAVITATIONAL


EquipmentIncluded:

• Gravitational Torsion Balance• support base with leveling feet• 1.5 kg lead balls (2)• plastic plate• replacement torsion ribbon

(part no. 004-06788)• 2-56 x 1/8 Phillips head screws (4)• Phillips screwdriver (not shown)

Additional Required:• laser light source (such as the PASCO OS-9171 He-Ne Laser)

• meter stick

Figure 2Equipment Included

1.5 kg leadmasses

replacementtorsion ribbon

plasticdemonstrationplatealuminum

plate

leveling sight

large massswivel support

pendulummirror

zero adjustknob

torsion ribbonhead

2-56x1/8Phillips headscrews

optical gradeglass window

leveling feet

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3

Attach to

Earth Ground.

GRAVITATIONAL


Equipment Parameters• Small lead balls

Mass: 38.3 g + 0.2 g (m2)Radius: 9.53 mmDistance from ball center to torsion axis: d = 50 .0 mm

• Large lead ballsMass: 1500 g + 10 g (m1)Radius: 31.9 mm

• Distance from the center of mass of the large ball to thecenter of mass of the small ball when the large ball isagainst the aluminum plate and the small ball is in thecenter position within the case: b = 46.5 mm (Toleranceswill vary depending on the accuracy of the horizontalalignment of the pendulum.)

• Distance from the surface of the mirror to the outer surfaceof the glass window: 11.4 mm

• Torsion Ribbon Material: Beryllium CopperLength: approx. 260 mmCross-section: .017 x .150 mm

Equipment SetupInitial Setup

1. Place the support base on a flat, stable table that is locatedsuch that the Gravitational Torsion Balance will be at least5 meters away from a wall or screen.

Note: For best results, use a very sturdy table, such as an opticstable.

2. Carefully remove the Gravitational Torsion Balance fromthe 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 thependulum chamber.

Note: Save the packing foam, and reinstall it each time theGravitational Torsion Balance is transported.

4. Fasten the clear plastic plate to the case with thethumbscrews.

IMPORTANT NOTES

➤ The Gravitational Torsion Balanceis a delicate instrument. Werecommend that you set it up in arelatively secure area where it is safefrom accidents and from those whodon’t fully appreciate delicateinstruments.

➤ The first time you set up thetorsion balance, do so in a placewhere you can leave it for at least oneday before attempting measurements,allowing time for the slight elongationof the torsion band that will occurinitially.

Keep the pendulum bob secured inthe locking mechanisms at all times,except while setting up andconducting experiments.

Do not touch the mirror on the pendulum.Figure 3Removing a plate from the chamber box

pendulumchamber

aluminum plate

pendulumbob

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4

Vertical Adjustment of the PendulumThe base of the pendulum should be flush with the floor of thependulum chamber. If it is not, adjust the height of the pendulum:

1. Grasp the torsion ribbon head andloosen the Phillips retaining screw(Figure 6a).

2. Adjust the height of the pendulumby moving the torsion ribbon headup or down so the base of thependulum is flush with the floor ofthe pendulum chamber(Figure 6b).

3. Tighten the retaining (Phillipshead) screw.

Leveling the Gravitational Torsion Balance1. Release the pendulum from the locking mechanism by

unscrewing the locking screws on the case, lowering thelocking mechanisms to their lowest positions (Figure 4).

Figure 5Using the leveling sight to level theGravitational Torsion Balance.

Look through thesight to view thereflection of thependulum bob inthe mirror.

Pendulum bobmust becentered overthe mirror.

pendulum

mirror

torsion ribbon

torsion ribbonhead

Figure 4Lowering the locking mechanism to release the pendulum bob arms

Turn lockingscrews clockwise.

lockingmechanisms

pendulumbob arm

2. Adjust the feet of the base until the pendulum is centered inthe leveling sight (Figure 5). (The base of the pendulum willappear as a dark circle surrounded by a ring of light).

3. Orient the Gravitational Torsion Balance so the mirror onthe pendulum bob faces a screen or wall that is at least 5meters away.

! " SIDE,CUTAWAYVIEW

"!

Figure 6Adjusting the height of the pendulum bob

The bottom of the pendulumbob should be flush with thefloor of the chamber.

Grasp the torsion ribbon headand loosen the Phillips screw.

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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). Toadjust them:

1. Mount a metric scale on the wall or other projection surfacethat is at least 5 meters away from the mirror of thependulum.

2. Replace the plastic cover with the aluminum cover.

3. Set up the laser so it will reflect from the mirror to theprojection surface where you will take your measurements(approximately 5 meters from the mirror). You will need topoint the laser so that it is tilted upward toward the mirrorand so the reflected beam projects onto the projection surface(Figure 8). There will also be a fainter beam projected off thesurface of the glass window.

Note: Vertical adjustment is only necessary at initial setup andwhen you change the torsion ribbon or if someone has loosenedthe retaining screw by mistake; it is not normally done duringeach experimental setup.

L

Figure 8Setting up the optical lever

SIDE VIEW

TOP VIEW

Llaser

reflected beam(from mirror)

screen with scale

Figure 7Aligning the pendulum bob rotationally

TOP, CUTAWAY VIEW

small mass

case plates

The pendulum bob armmust be centeredrotationally between theplates.

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6

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3. Rotationally align the case by rotating it until the laser beamprojected from the glass window is centered on the metricscale (Figure 9).

4. Rotationally align the pendulum arm:

a. Raise the locking mechanisms by turning the lockingscrews until both of the locking mechanisms barelytouch the pendulum arm. Maintain this position for afew moments until the oscillating energy of thependulum is dampened.

b. Carefully lower the locking mechanisms slightly so thependulum can swing freely. If necessary, repeat thedampening exercise to calm any wild oscillations of thependulum bob.

c. Observe the laser beam reflected from the mirror. In theoptimally aligned system, the equilibrium point of theoscillations of the beam reflected from the mirror will bevertically aligned below the beam reflected from theglass surface of the case (Figure 9).

d. If the spots on the projection surface (the laser beamreflections) are not aligned vertically, loosen the zeroadjust thumbscrew, turn the zero adjust knob slightly torefine the rotational alignment of the pendulum bobarms (Figure 10), and wait until the movement of thependulum stops or nearly stops.

e. Repeat steps 4a – 4c as necessary until the spots arealigned vertically on the projection surface.

5. When the rotational alignment is complete, carefully tightenthe zero adjust thumbscrew, being careful to avoid jarring thesystem.

Figure 9Ideal rotational alignment (zeroing) of thependulum

location of theprojected laserbeam from the glasswindow

zero adjust thumbscrew

zero adjust knob

Figure 10Refining the rotational alignment of thependulum bob

Hints for speedier rotational alignments:• Dampen any wild oscillations of the pendulum bob with the

locking mechanisms, as described;

• Adjust the rotational alignment of the pendulum bob usingsmall, smooth adjustments of the zero adjust knob;

• Exercise patience and finesse in your movements.

location of theprojected laser beamfrom the mirror

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7

Attach to

Earth Ground.

GRAVIT

TORSION B

Setting up for the Experiment1. Take an accurate measurement of the distance from the

mirror to the zero point on the scale on the projectionsurface (L) (Figure 8). (The distance from the mirror surfaceto 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), andground it to the earth.

3. Place the large lead masses on the support arm, and rotatethe arm to Position I (Figure 12), taking care to avoidbumping 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 ofthree methods: the final deflection method, the equilibriummethod, or the acceleration method.

Note: The pendulum may require several hours to reach restingequilibrium. To shorten the time required, dampen the oscillationof 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 11Attaching the grounding strap to thegrounding screw

copper wire toearth ground

groundingscrew

Figure 12

Large Masses:Position I

Large Masses:Position II

Mirror

Light beam

Measuring the GravitationalConstantOverview of the Experiment

The gravitational attraction between a 15 gram mass and a 1.5 kgmass when their centers are separated by a distance ofapproximately 46.5 mm (a situation similar to that of theGravitational Torsion Balance ) is about 7 x 10 -10 newtons. If thisdoesn’t seem like a small quantity to measure, consider that theweight of the small mass is more than two hundred million timesthis amount.

case

glasswindow

smallmass

Moving the large masses into Position I


8

d

b

The enormous strength of the Earth's attraction for the smallmasses, in comparison with their attraction for the large masses, iswhat originally made the measurement of the gravitationalconstant such a difficult task. The torsion balance (invented byCharles Coulomb) provides a means of negating the otherwiseoverwhelming effects of the Earth's attraction in this experiment.It also provides a force delicate enough to counterbalance the tinygravitational force that exists between the large and small masses.This force is provided by twisting a very thin beryllium copperribbon.

The large masses are first arranged in Position I, as shown inFigure 12, and the balance is allowed to come to equilibrium. Theswivel support that holds the large masses is then rotated, so thelarge masses are moved to Position II, forcing the system intodisequilibrium. The resulting oscillatory rotation of the system isthen observed by watching the movement of the light spot on thescale, as the light beam is deflected by the mirror.

Any of three methods can be used to determine the gravitationalconstant, G, from the motion of the small masses. In Method I,the final deflection method, the motion is allowed to come toresting equilibrium—a process that requires several hours—andthe result is accurate to within approximately 5%. In method II,the equilibrium method, the experiment takes 90 minutes or moreand produces an accuracy of approximately 5% when graphicalanalysis is used in the procedure. In Method III, the accelerationmethod, the motion is observed for only 5 minutes, and the resultis accurate to within approximately 15%.

METHOD I: Measurement by Final DeflectionSetup Time: ~ 45 minutes; Experiment Time: several hoursAccuracy: ~ 5%

TheoryWith the large masses in Position I (Figure 13), the gravitationalattraction, F, between each small mass (m2) and its neighboringlarge mass (m1) is given by the law of universal gravitation:

F = Gm1m2 /b2 (1.1)

where b = the distance between the centers of the twomasses.

Large Masses:Position I

Large Masses:Position II

Figure 13Origin of variables b and d

➤ Note: 5% accuracy is possible inMethod I if the experiment is set up on asturdy table in an isolated locationwhere it will not be disturbed byvibration or air movement.

➤ Note: 5% accuracy is possible inMethod II if the resting equilibriumpoints are determined using a graphicalanalysis program.

012–06802B Gravitational Torsion Balance

9

The gravitational attraction between the two small masses andtheir neighboring large masses produces a net torque ( grav) onthe system:

grav = 2Fd (1.2)

where d is the length of the lever arm of thependulum bob crosspiece.

Since the system is in equilibrium, the twisted torsion bandmust be supplying an equal and opposite torque. This torque( band) is equal to the torsion constant for the band (κ) times theangle through which it is twisted (θ), or:

band = –κθ. (1.3)

Combining equations 1.1, 1.2, and 1.3, and taking into accountthat 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 inequation 1.4 — it is necessary to observe the oscillations of thesmall mass system when the equilibrium is disturbed. Todisturb the equilibrium (from S1), the swivel support is rotatedso the large masses are moved to Position II. The system willthen oscillate until it finally slows down and comes to rest at anew equilibrium position (S2) (Figure 14).

At the new equilibrium position S2, the torsion wire will still betwisted through an angle θ, but in the opposite direction of itstwist in Position I, so the total change in angle is equal to 2θ.Taking into account that the angle is also doubled uponreflection from the mirror (Figure 15):

∆S = S2 – S1,

4θ = ∆S/L or

θ = ∆S/4L (1.5)

T20

60

S (c

m)

Time (min)

s2

s1

Figure 14Graph of Small Mass Oscillations

G = κθb22dm1m2

S1 S2

∆S2

θ

L

Figure 15Diagram of the experiment showing theoptical lever.

2θ» tan (2θ) = ∆S2L

Position II

Position I


10

The torsion constant can be determined by observing the period(T ) of the oscillations, and then using the equation:

T2 = 4π2I/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 thesmall masses is negligibly small compared to that of themasses 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 ormeasurable:

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 (∆S) and theperiod of oscillation (T ), the value of G can therefore bedetermined.

Procedure1. Once the steps for leveling, aligning, and setup have been

completed (with the large masses in Position I), allow thependulum to stop oscillating.

2. Turn on the laser and observe the Position I end point ofthe balance for several minutes to be sure the system is atequilibrium. Record the Position I end point (S1) asaccurately as possible, and indicate any variation overtime as part of your margin of error in the measurement.

G = π 2∆Sb 2 (d2 + 2 52 5r 2)T 2m 1Ld

I = 2m2(d 2 + 2 52 5r2)

κ = 8π2m2d 2 + 2 52 5r2

T2


11

3. Carefully rotate the swivel support so that the largemasses are moved to Position II. The spheres should bejust touching the case, but take care to avoid knocking thecase and disturbing the system.

Note: You can reduce the amount of time the pendulumrequires to move to equilibrium by moving the largemasses in a two-step process: first move the large massesand support to an intermediate position that is in themidpoint of the total arc (Figure 16), and wait until thelight beam has moved as far as it will go in the period;then move the sphere across the second half of the arcuntil the large mass support just touches the case. Use aslow, smooth motion, and avoid hitting the case whenmoving the mass support.

4. Immediately after rotating the swivel support, observe thelight spot and record its position (S1).

5. Use a stop watch to determine the time required for oneperiod of oscillation (T). For greater accuracy, includeseveral periods, and then find the average time requiredfor one period of oscillation.

Note: The accuracy of this period value (T ) is veryimportant, since the T is squared in the calculation of G.

6. Wait until the oscillations stop, and record the restingequilibrium point (S2).

Analysis1. Use your results and equation 1.9 to determine the value

of G.

2. The value calculated in step 2 is subject to the followingsystematic error. The small sphere is attracted not only toits neighboring large sphere, but also to the more distantlarge sphere, though with a much smaller force. The

S1 S2

L

Figure 16Two-step process of moving the largemasses to reduce the time required to stoposcillating

Position I Position II

intermediateposition


12

geometry for this second force is shown in Figure 17 (thevector arrows shown are not proportional to the actualforces).

From Figure 17,

The force, F0 is given by the gravitational law, whichtranslates, in this case, to:

and has a component ƒ that is opposite to the direction ofthe force F :

This equation defines a dimensionless parameter, b, that isequal to the ratio of the magnitude of ƒ to that of F. Usingthe equation F = Gm1m2/b

2, 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 smallsphere from both large masses, and F is the force ofattraction to the nearest large mass only.

Similarly,G = G0(1 - b)

where G is your experimentally determined value for thegravitational constant, and G0 is corrected to account forthe systematic error.

Finally, G0 = G/(1 - b)

Use this equation with equation 1.9 to adjust yourmeasured value.

F0 =Gm2m1(b2 + 4d 2)

f = Gm2m1b

(b2 + 4d 2)(b2 + 4d 2)12= βF

Φ

d

bF0

F

f

Figure 17Correcting the measured value of G

sinΦ = b(b2 + 4d 2)

1 21 2

f = F0 sinΦ


13

METHOD II: Measurement by EquilibriumPositions

Observation Time: ~ 90+ minutesAccuracy: ~ 5 %

TheoryWhen the large masses are placed on the swivel support andmoved to either Position I or Position II, the torsion balanceoscillates for a time before coming to rest at a new equilibriumposition. This oscillation can be described by a damped sinewave with an offset, where the value of the offset representsthe equilibrium point for the balance. By finding theequilibrium point for both Position I and Position II and takingthe difference, the value of ∆S can be obtained. The remainderof the theory is identical to that described inMethod I.

Procedure1. Set up the experiment following steps 1–3 of Method I.

2. Immediately after rotating the swivel support to PositionII, observe the light spot. Record the position of the lightspot (S) and the time (t) every 15 seconds. Continuerecording the position and time for about 45 minutes.

3. Rotate the swivel support to Position I. Repeat theprocedure described in step 2.

Note: Although it is not imperative that step 3 be performedimmediately after step 2, it is a good idea to proceed withit as soon as possible in order to minimize the risk that thesystem will be disturbed between the two measurements.Waiting more than a day to perform step 3 is not advised.

Analysis1. Construct a graph of light spot position versus time for

both Position I and Position II. You will now have agraph similar to Figure 18.

2. Find the equilibrium point for each configuration byanalyzing the corresponding graphs using graphicalanalysis to extrapolate the resting equilibrium points S1and S2 (the equilibrium point will be the center line aboutwhich the oscillation occurs). Find the difference between

60

S (c

m)

Time (min)

S2

S1

0

20

Figure 18Typical pendulum oscillation patternshowing equilibrium positions

Note: To obtain an accuracy of 5%with this method, it is importantto use graphical analysis of theposition and time data toextrapolate the resting equilibriumpositions, S1 and S2.


14

the two equilibrium positions and record the result as ∆S.

3. Determine the period of the oscillations of the small masssystem by analyzing the two graphs. Each graph willproduce a slightly different result. Average these resultsand record the answer as T.

4. Use your results and equation 1.9 to determine the valueof G.

5. The value calculated in step 4 is subject to the samesystematic error as described in Method I. Perform thecorrection procedure described in that section (Analysis,step 3) to find the value of G0.

METHOD III:Measurement by Acceleration

Observation Time: ~ 5 minutesAccuracy: ~ 15%

TheoryWith the large masses in Position I, the gravitational attraction,F, between each small mass (m2) and its neighboring largemass (m1) is given by the law of universal gravitation:

F = Gm1m2 /b2 (3.1)

This force is balanced by a torque from the twisted torsionribbon, so that the system is in equilibrium. The angle of twist,θ, is measured by noting the position of the light spot wherethe reflected beam strikes the scale. This position is carefullynoted, and then the large masses are moved to Position II. Theposition change of the large masses disturbs the equilibrium ofthe system, which will now oscillate until friction slows itdown to a new equilibrium position.

Since the period of oscillation of the small masses is long(approximately 10 minutes), they do not move significantlywhen the large masses are first moved from Position I toPosition II. Because of the symmetry of the setup, the largemasses exert the same gravitational force on the small massesas they did in Position I, but now in the opposite direction.Since the equilibrating force from the torsion band has not


15

changed, the total force (Ftotal) that is now acting to acceleratethe small masses is equal to twice the original gravitationalforce from the large masses, or:

Ftotal = 2F = 2Gm1m2/b2 (3.2)

Each small mass is therefore accelerated toward itsneighboring large mass, with an initial acceleration (a0) that isexpressed in the equation:

m2a0 = 2Gm1 m2/b2 (3.3)

Of course, as the small masses begin to move, the torsionribbon becomes more and more relaxed so that the forcedecreases and their acceleration is reduced. If the system isobserved over a relatively long period of time, as in Method I,it will be seen to oscillate. If, however, the acceleration of thesmall masses can be measured before the torque from thetorsion ribbon changes appreciably, equation 3.3 can be used todetermine G. Given the nature of the motion—dampedharmonic—the initial acceleration is constant to within about5% in the first one tenth of an oscillation. Reasonably goodresults can therefore be obtained if the acceleration is measuredin the first minute after rearranging the large masses, and thefollowing relationship is used:

G = b2a0 /2m1 (3.4)

The acceleration is measured by observing the displacement ofthe light spot on the screen. If, as is shown in Figure 19:

∆s = the linear displacement of the small masses,

d = the distance from the center of mass of the smallmasses to the axis of rotation of the torsionbalance,

∆S = the displacement of the light spot on the screen,and

L = the distance of the scale from the mirror of thebalance,

then, taking into account the doubling of the angle onreflection,

∆S = ∆s(2L/d ) (3.5)

Using the equation of motion for an object with a constant

S1 S2

∆S2

2θL

Figure 19Source of data for calculations inMethod III

()*+,-*-,./*012.)3,./1'*0*/45 !"#6!%&!#'

16

acceleration (x = 1/2 at2), the acceleration can be calculated:

a0 = 2∆s/t2 = ∆Sd/t2L (3.6)

By monitoring the motion of the light spot over time, theacceleration can be determined using equation 3.6, and thegravitational constant can then be determined using equation3.4.

Procedure1. Begin the experiment by completing steps 1–3 of the

procedure detailed in Method I.

2. Immediately after rotating the swivel support, observe thelight spot. Record the position of the light spot (S) and thetime (t) every 15 seconds for about two minutes.

Analysis1. Construct a graph of light spot displacement

(∆S = S - S1) versus time squared (t2), with t2 on thehorizontal axis (Figure 20). Draw a best-fit line throughthe observed data points over the first minute ofobservation.

2. Determine the slope of your best-fit line.

3. Use equations 3.4 and 3.6 to determine the gravitationalconstant.

4. The value calculated in step 3 is subject to a systematicerror. The small sphere is attracted not only to itsneighboring large sphere, but also to the more distantlarge sphere, although with a much smaller force. Use theprocedure detailed in Method I (Analysis, step 3) tocorrect for this force.

∆S

t2 (sec2)

0

1

2

3

4

5

6

7

8

9

225 2025900 3600 5625 8100 11025

BEST FIT LINE

CURVETHROUGH

DATA

Figure 20Sample data and best-fit line

!"#6!%&!#' ()*+,-*-,./*012.)3,./1'*0*/45

17

MaintenanceReplacing the Torsion Ribbon Assembly

If the torsion ribbon breaks, replace it as follows:

1. Remove the plates, and raise the locking mechanism usingthe locking screws until the pendulum arms are securelyanchored (Figure 21a).

2. Grasp the pendulum bob near the bottom ribbon tab tostabilize it.

3. Loosen the Phillips screw on the bottom tab of the torsionribbon assembly (Figure 21a), and remove the bottom halfof the broken ribbon assembly.

4. Loosen the Phillips screw at the top of the balanceassembly (Figure 21b).

5. Grasp the torsion ribbon head and remove the top portionof the broken torsion ribbon assembly.

6. Attach the top tab of the new torsion ribbon to the torsionribbon head using the Phillips screw, being sure thecopper disc on the tab is in contact with the torsion ribbonhead (Figure 22). Align the tab with the face of the torsionribbon head.

➀ Turn locking screws untillocking mechanism anchorsthe pendulum arms.

➂ Loosenthe Phillipsscrew.

lockingmechanism

brokentorsionribbon

➁ Grasp thependulum bob hereto stabilize it.

➃ Loosen thePhillips screw.

Figure 21Securing the pendulum bob beforeremoving a broken torsion ribbon, andloosening the torsion ribbon head

➄ Grasp the torsionribbon head and removethe top portion of thebroken ribbon assembly.

The copperdisk mustcontact thetorsionribbonhead.

Attach the tab withthe Phillips screw.

torsionribbon

ribbon tab

torsion ribbon head

Figure 22Attaching the top tab of the torsion ribbon assembly to the torsionribbon head

"

!

7. Thread the ribbon through the shaft.

8. Using the zero adjust knob, align the bottom tab with theface of the pendulum bob.

zero adjustknob


18

Note: Be sure the ribbon is not twisted.

9. Tighten the Phillips screw on the top of the balance tosecure the torsion ribbon head.

10. Attach the bottom tab of the ribbon to the pendulum bobusing the Phillips screw.

11. Replace the back plate.

12. Level and align the pendulum according to the instructionsin the Equipment Setup section of this manual.

Transporting and Storing1. To prepare the Gravitational Torsion Balance for

transporting or storing:

a. Remove the front plate.

b. Raise the locking mechanism to securely anchorthe pendulum bob.

c. Check to be sure that the torsion ribbon ishanging straight down the center of the tube. If itis not, lower the locking mechanisms, be sure thetorsion wire is centered, and raise the lockingmechanisms again. Repeat as necessary until theribbon is centered in the tube.

d. Reinstall the packing foam into the chamber tosecure the pendulum bob.

e. Replace the plate.

2. The Gravitational Torsion Balance may be stored flat inits shipping container.

3. Store in a cool, dry place, and protect the device from anyjarring or rough handling.

Safety PrecautionThe small and large masses are made of lead, which is toxic ifingested. Use appropriate precautions for handling lead,including hand washing after handling the masses.

Note: To avoid breaking thetorsion ribbon, the lockingmechanism must be fully raisedon both sides during anymoving or transporting of theGravitational Torsion Balance.


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Technical Support

FeedbackIf you have any comments about the product ormanual, please let us know. If you have anysuggestions on alternate experiments or find aproblem in the manual, please tell us. PASCOappreciates any customer feedback. Your inputhelps us evaluate and improve our product.

To Reach PASCOFor technical support, call us at 1-800-772-8700(toll-free within the U.S.) or (916) 786-3800.

fax: (916) 786-3292

e-mail: [email protected]

web: www.pasco.com

Contacting Technical SupportBefore you call the PASCO Technical Support staff,it would be helpful to prepare the followinginformation:

➤ If your problem is with the PASCO apparatus,note:

- Title and model number (usually listed on thelabel);

- Approximate age of apparatus;

- A detailed description of the problem/sequenceof events (in case you can’t call PASCO rightaway, you won’t lose valuable data);

- If possible, have the apparatus within reachwhen calling to facilitate description ofindividual parts.

➤ If your problem relates to the instruction manual,note:

- Part number and revision (listed by month andyear on the front cover);

- Have the manual at hand to discuss yourquestions.

WPI Physics Dept. Intermediate Lab 2651 Cavendish...

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