Universal Gravitational Constant EX5550 Page of
1Universal Gravitational Constant
Equipment
1

Gravitational Torsion Balance

AP8215A

1

XY Adjustable Diode Laser

OS8526A

1

45 cm Steel Rod

ME8736

1

Large Rod Base

ME9735

1

Aluminum Table Clamp

ME8995

1

Small “A” Base

ME8976

1

USB Camera Microscope

PS2343

1

Flex Rod

ME8978A

1

Polarizer Set

OS8473

1

Sensor Handle


1

Double Rod Clamp


1

¼ 20 plastic thumb nut



Needed but not included


1

Transparent ruler


1

Meter Stick

SE7333

1

Digital Calipers

SE8710


PASCO Capstone

UI5400


Optional but very useful


1

Antivibration Pads (set of 3)**


1

Pulley Mounting Rod (use with pads)

SA9242

** Antivibration pads are designed to use with a telescope and are available from Celestron, Meade, or Orion Telescopes for $50$70. Look online. Using the pads decreases the amount of vibrations that the Torsion Balance picks up from the surroundings.
Introduction
The 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. Data collection requires less than 45 minutes. The data may then be transferred to each lab group using flash drives. This makes it possible to perform this classic experiment in a normal laboratory period with an accuracy of better than 3%.
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.
Historical Overview
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.
Newton’s law of universal gravitation:
F = GmM/r^{2}
where m and M are the masses of the objects, r is the distance between them, and
G = 6.67 x 10^{11} Nm^{2}/kg^{2}
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 G allowed 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.
The gravitational attraction between a 38 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 1.8 x 10^{9} 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.
T he 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.
Cavendish Balance: the large masses are first arranged in Position I, as shown in Figure 1, 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.
Theory
Newton’s law of universal gravitation:
F = GmM/b^{2} Eq. (1)
w here m and M are the masses of two spherical objects, b is the centertocenter distance between them, F is the force of attraction between the two masses, and G is a constant to be measured in this experiment. The force (F) is extremely small (< 2 nN) and will be measured using a very sensitive torsion balance. The torsion balance consists of two tungsten spheres, each of mass m = 38.5 g and diameter = 1.635(5) cm, attached to a thin aluminum rod that yields a centertocenter distance for the two balls of d =10.0(1) cm. The small ball dumbbell is suspended by a 40 cm beryllium copper ribbon. The large balls are made of tungsten and each have a mass of M = 1.500(5) kg and diameter D.
When the two large masses (M) are in the center position (or removed), if the system is perfectly aligned, the two small masses (m) will be in the position shown in black in Figure 2 so that they are parallel to the screen. The reflection from the laser would then fall at point 2 on the screen. We will assume this is the case since it makes the derivation easier. If the small masses differ from parallel by less than 0.1 radians (5^{o}), the errors introduced (by assuming tan = _{(rad)}) are less than 0.5%. When the large masses (M) are moved to the position shown in Figure 2, the gravitational force (F) causes a torque () about the center of the dumbbell formed by the small masses and they rotate through an angle to the equilibrium position shown in blue. The spot on the wall due to the laser will shift by an angle of 2 or a linear distance of S/2. We then reverse the positions of the large masses and the laser spot moves to position 1. From symmetry, the distance between 1&2 is the same as between 2&3, so the total distance between 1&3 is S and this is what we will actually measure.
T
he torsional pendulum obeys Hooke’s Law for a rotating system, so
= C Eq. (2)
Where C is a constant determined by the beryllium copper ribbon supporting the small mass dumbbell.
= 2(d/2)F cos Eq. (3)
Actually, the angle is slightly greater than because F is along the line of centers and is not quite vertical (on the diagram). However, since with the screen at 1.5 m, S is less than 4 cm, is less that 1/2^{o} and we may approximate this as
= dF = C Eq. (4)
How large a percent error does this approximation introduce? See Appendix 6 for a discussion of small corrections.
The torsional pendulum will overshoot the equilibrium position and oscillate for several hours since there is very little damping in the system. Since it obeys Hooke’s law, the motion is simple harmonic (slightly damped) and the angular frequency is given by
= (C/I)^{½} Eq. (5)
where I is the rotational inertia for the small mass dumbbell about its center. Using the parallel axis theorem,
I = 2[(2/5)mr^{2} + m(d/2)^{2}] = m(4r^{2}/5 + d^{2}/2) Eq. (6)
Equations 5&6 yield
C = ^{2}m(4r^{2}/5 + d^{2}/2) Eq. (7)
And Equation 4 becomes
dGMm/b^{2} = ^{2}m(4r^{2}/5 + d^{2}/2) Eq. (8)
Solving for G yields
G = ^{2}b^{2}(4r^{2}/5 + d^{2}/2)/Md Eq. (9)
Setup
Do Not Touch the Cavendish Balance! Hours are required for unwanted oscillations to damp out!
Do Not Touch the Adjustment Wheels at the top of the Cavendish Balance!!! Many hours are required to correctly adjust them.
Note to instructor: it is nice to put the plastic plate on so the students can see the small balls inside, but do it after data collection is over for all the labs using the balance. Do not do the experiment with the plastic plate since small electrostatic charges can interfere.
System should be similar to Figure 3. The laser spot should move along the top of the transparent ruler.
Figure 3: Setup
Procedure

In PASCO Capstone, create a Movie Display by dragging the icon from the right palette to the page. Click on “Record Movie with Synched Data” and select the camera you are using.

Click on the Properties icon in the movie toolbar.

Click on Movie Recording. Set the Frame Rate as low as possible. It should be no more than 5 (frames per second) and use 1 fps if possible. Set Video Compression to DV Video Encoder if using a PC or to 240 H264 if using a MAC.

Click PREVIEW (the red circle above the video screen). You want to position the webcam so that the entire motion of the laser spot will be magnified as much as possible without moving off the webcam screen. To do this you need to understand a bit about how the system will work. If the Cavendish mirror is 1.5 m from the screen, then the laser spot equilibrium position when the balls are in their clockwise (seen from above – see Figure 1) position will be nearly 4.0 cm to the left of the equilibrium position when the balls are in their counterclockwise position. For example, if the equilibrium position of the laser spot is at the 15 cm mark on the ruler when the balls are in the center position, then it will shift to about the 13 cm mark when the balls are clockwise and the 17 cm mark when the balls are counterclockwise. This should allow you to center the webcam on the expected equilibrium position. The amplitude of the laser spot’s motion will be determined by the position of the spot when you shift the balls. It is easiest to predict what will happen if you move the balls when the spot is not moving, either because it is at equilibrium or is near a turning point since then the balls have no kinetic energy. For example, if the balls are in the center position with the spot in equilibrium at 15 cm when you shift the balls clockwise, the spot will then oscillated back and forth roughly between 11 cm and 15 cm (around the equilibrium at 13 cm). If you let the oscillation damp down until it is between 12 cm and 14 cm, then if you move the balls to the counterclockwise position when the spot is near the 14 cm turning point, the new oscillation will be between 14 cm and 20 cm (around the equilibrium at 17 cm). This should allow you to figure out where to position the webcam for the best results. Note that if you get it wrong and the spot goes off screen, you can simply reposition the webcam, delete the run, and start the recording over.

Adjust the webcam so the image is focused sharply, the ruler image is parallel to the top of the screen, and the path of the laser spot along the top of the ruler is across the center of the screen to minimize “keystoning”.

If necessary, adjust the polarizer to prevent the laser spot from “blooming” on the screen, but be careful not to cause vibrations in the system.

With the balls in either the clockwise or the counterclockwise position and at least one of the balls touching the case and the laser spot moving at least 3 cm between turning points (see the discussion in step 3 above to adjust the motion of the spot), click RECORD and let it run for 17 minutes (two periods) before clicking STOP. The period of oscillation is about 8 minutes. Note: much less than two periods begins to degrade the data significantly, but there is little gain in going for more than two periods.

To get the second position, when the laser spot is near the turning point nearest where the new equilibrium will be (see Step 3 above), carefully rotate the balls to the other position near the case. Do this as gently as possible. At least one ball should touch the case, but try to do this very, very gently.

Click PREVIEW and move the webcam to the appropriate position to record the new motion.

Click RECORD and let it run for 17 minutes before clicking STOP.

Create a table like the one shown below.

After completing the two movies determine:

The average diameter and uncertainty for each of the two 1.5 kg balls. Record in the table. Use calipers if available.

The width of the box containing the small ball dumbbell. Use calipers if available. Estimate the uncertainty. Record.

The perpendicular distance to the screen to the mirror attached to the small mass dumbbell. To do this, measure from the screen to the front of the box by the mirror and add 1.1 cm to include the distance from the front of the box to the mirror. Estimate the uncertainty. Record.

Calculate the centertocenter distance (b) for the small ball to big ball. Note if either ball fails to touch the case when the other ball is against the case and estimate the gap width if there is one. Include this in the uncertainty (b).

Save. At this point, data can be transferred to each lab group using a flash drive or other data transfer method.
Analysis

Click on Record Movie with Synced Data. Select USB Video Device (IceCam 2) and click OK. Do not go back to the Procedure page! It could cause the program to lose your data.

Click on the Properties in the movie toolbar and set Movie Playback for x16 and a frame rate of 5 or 1 depending on what you used in record but do not set the frame rate if you are using a MAC or you will lose your data!

Click the green Playback Mode icon at the lower left of the screen. Select Run #1.

Set the movie to its beginning and then click on the play button. Observe the first three turning points and note their positions.

Average the 1^{st} and 3^{rd} turning points and then average that value with the 2^{nd} turning point. This will give an approximate value for the zero position. Round off to the nearest millimeter. Enter the value as Initial 1.

Repeat steps 35 except this time select Run #2 and record the value as Initial 2.

Select Run #1 (at the bottom of the screen). Click on the Video Analysis button at the left of the movie toolbar.

Drag the caliper points of the Calibration tool to two points on the clear ruler near the region where the laser spot moves. Note the distance between the two points.

Click the Properties icon in the toolbar, select Calibration Tool and in the popup box set the Real World Length to the value noted in step 6 (probably around 3 or 4 cm).

Click the Calibration Tool in the toolbar to hide the Calibration Tool.

Drag and rotate the coordinate axes so that the coordinate zero is at the Initial 1 position noted in step 5 and the horizontal axis is along the path of the laser spot (probably the edge of the ruler.)

Click on the Coordinate Axes icon in the toolbar to hide the coordinate axes.

The frame advance button (green at the left of the toolbar) should already be depressed.

Click on the Properties button in the toolbar, select Overlay and set the Frame Increment so you will look at frames that are 10 s apart. For a PC this means set Frame Increment to 50 if you recorded at 5 frames per second or at 10 if you recorded at 1 fps. For MAC select a Frame Increment of 300 for all cases since MAC assumes 30 fps (MAC actually fills in the missing frames by copying the ones that were recorded.)

Click on the screen below the laser spot. A red cross will appear. You want the crosses you make to be below (or above) where the spot moves so the crosses will not hide the spot on subsequent oscillations. If the crosses interfere with the data you can turn them off. Click on Properties, Tracked Object, and click Visible off.

Repeat step 15 until you reach the end of the video.

Repeat steps 716 for Run #2.
Curve Fit

Create a graph of “xposition tracked object 2” vs. time.

Click the Select Measurement button on the vertical axis of the graph and select “xposition tracked object 2”. Select the first of the 17 minute movies (Run #1) using the black triangle by the Run Select icon in the graph toolbar.

Click on the black triangle by the Curve Fit icon and select Damped Sine.

Click on the Damped Sine box and the Curve Fit Editor will appear at the left of the screen.

Click on the Curve Fit Editor. Set B to zero and Lock it. Click Update Fit. You should get a sine curve that roughly fits the data.

Unlock B and click Update Fit. You should now have a damped sine curve that fits the data. Close the Curve Fit Editor.

Right click in the Damped Sine box. Select Curve Fit Properties, Numerical Format, Coefficients and change the Number of Significant Figures to 4.

Record the values for the angular frequency () and the correction to the zero position (C) in the table.

To estimate the uncertainties in and C, click on the Highlight range icon and adjust the handles on the box to include 600 seconds of data. Slide the box left and right to see how much and C change. From the range of values for and C, estimate the uncertainties in and C. Enter them in the table as and C.

Note that the C value is the correction to the Initial 1 value from the previous page. Calculate the corrected zero value and enter it and its uncertainty as Position 1 and 1. Note that this should include some uncertainty in how well the Coordinate origin could be placed.

Repeat for Run #2. Determine Position 2 and 2.

From the zero positions, determine the distance the spot moved S and its uncertainty (S). Record.

Enter the best value of and using both 17minute movie values.
Calculation of G
From the L a S values in the table, calculate and its uncertainty see Figure 4. Enter the values in the table. Use the values in the table to calculate G using Equation 9 and enter your value in the table.
G = ^{2}b^{2}(4r^{2}/5 + d^{2}/2)/Md Eq. (9)
Uncertainty in G
For a product or quotient of terms of the form ABn/C = D where the uncertainties in A, B, and C are known, the uncertainty (D) in D is given by:
D/D = [(A/A)^{2} + (nB/B)^{2} + (C/C)^{2}]^{1/2} Eq. (10)
Since the r^{2 }term is much smaller than the d^{2} term in Equation 9, it may be ignored. The uncertainty in M can also be ignored since M/M is small compared to the other terms. Equation 10 then yields
G/G = [()^{2} + (2)^{2} + (2b/b)^{2} + (d/d)^{2}]^{1/2} Eq. (11)
Calculate the uncertainty in G and enter it in the table.
Corrections to G
The value you obtained for G is probably lower than the actual value by more than the uncertainty because we have ignored two significant errors.

The actual centertocenter distance between the balls is less than the b distance we used since the small balls move toward the large balls, decreasing the distance by b. Calculate Δb from θ and d (see Figure 4) and enter the value in the Table 8. Calculate "b correct" and record it. You correct the G value by multiplying by (b correct/b)². This should decrease G by around 12%.

There is a small counter torque on the small balls produced by the force (f) that the large ball exerts on the opposite small ball. Instead of
= F d Eq. (4)
the true torque is
= d(Ff sin φ) Eq. (12)
and the true value of G should be larger than the value you calculated by a factor of
factor = F/(Ff sin f) = 1/[1 (f/F)sin φ] =1/[1 ([GmM/(b²+d²)]/[GmM/b²])sin φ]
= 1/[1 (b²/(b²+d²))(b/(b²+d²)^{1/2}) = 1/(1 [b/(b²+d²)^{1/2}]³) Eq. (14)
Calculate the (total) correction factor from step 1 and equation 14 and enter it in the table. Correct your value for G and enter it in the table. Does your value for G +/ G agree with that given in your textbook? We have ignored the rotational inertia of the aluminum rod connecting the two small balls. How would this affect our value for G? The rod is about 3 mm in diameter and 10 cm long and has a density of 2.7 g/cm^{3}. Would it have a measurable effect? We have also ignored the gravitational attraction of the case for the small balls and the attraction of the large balls for the aluminum rod between the small masses. How would this affect our value for G?














































Corrections to Rotational Inertia due to Al rod
I_{true} = 2m(d/2)² + 2(2/5)mr² + (1/12)m_{r}d²
where m_{r} =
I_{true} /[2m(d/2)²+ 2(2/5)mr²]= 1 + (1/6)m_{r}/m/[1 + (8/5)(r/d)²]
Correction due to attraction of Al rod
m_{r} =
s_{m} =
dτ_{r} = 2 dF sin β (d/2  x) = 2(GM dm/s²)(b/s)(d/2  x)
= 2(GMbm_{r}/d)(dx/s³)(d/2  x)
τ_{r} = 2(GMm_{r}b/d)(dx/s³)(d/2  x) from x=0 to x=d
τ_{r}/τ = (m_{r}/m)(b/s_{m} + 2b³/d²s_{m}  2b²/d²)
Correction due to the aluminum holder for the large masses
The aluminum holder consists of two flat doughnuts connected by a flat bar. Each doughnut has an outer diameter of 5.00 cm, and inner diameter of 3.25 cm and a thickness of 0.80 cm. In addition, the adaptor ring has a mass of 6.5 g. The bar has a negligible affect since it has less mass and is more or less directly below the case so produces very little torque. The mass of the doughnut ring is:
m_{ring} =
In addition, the aluminum holder is below the horizontal plane determined by the centers of the balls by about 3.3 cm. The horizontal distance from m to the center of the ring is b. Treating the mass of the doughnut as if it were all at the center of the (not a very good approximation), the distances to the ring and bar are:
d_{ring} =
F = the force between m and M = GmM/b^{2}
F_{ring}/F =
The ratio of the force due to the ring the torque due to the doughnut is reduced by cos where is the angle the force is below the plane of the masses and tan = 3.5 cm/b = 3.5/4.27, so = 39^{0} and cos and the ratio of the torque produced by the ring to the torque produced by the large mass, M, is how much?
This means we can ignore the effect of the bar but not the effect of the doughnut. The 1% we derived for the doughnut should be regarded as an upper bound and probably rather high due to the assumption that we could treat it as a point mass. The reason it is high is that the part of the ring that feels the greatest force is also most nearly below the case so its horizontal component is decreased the most.
This can be directly verified by performing the experiment without the large masses in place. The data for this is included in the file called “ring only”. The results of analyzing that data is
S_{ring}/(S – S_{ring}) =
Appendix I: Initial SetupThis should be done at least four hours before the laboratory to allow time for unwanted oscillations to damp out. If you are setting up the Cavendish balance for the first time, you should begin the setup weeks in advance!
Preliminary Set Up (Note: be sure the thumbscrews on the bottom of the pendulum chamber are in the up position so the pendulum bob dumbbell is pushed up against the case (see Figure 6) and there is no tension in the beryllium copper ribbon before you move the unit! Breaking the ribbon is not a good thing!)

Place the support base on a flat, stable table that is located such that the Gravitational Torsion Balance will be about 1.5 meters away from a wall or screen. A greater distance may be used, but does not improve precision since the laser spot gets larger. If you use a larger distance, you may need to increase the distance of the video camera from the screen as well. For best results, use a very sturdy table, such as an optics table.

It may be necessary to put a short rod in the third hole to allow the leveling screws to level the system. The short rod is essential if you use the recommended antivibration pads.

If you have the recommended antivibration pads, place one under each of the three contact points.

Carefully secure the Gravitational Torsion Balance in the base.

Remove the front plate by removing the thumbscrews.

Fasten the clear plastic plate to the case with the thumbscrews.
Appendix II. Leveling the Gravitational Torsion Balance

Release the pendulum from the locking mechanism by unscrewing the locking screws on the case, lowering the locking mechanisms to their lowest positions (Figure 6).

Adjust the feet of the base until the pendulum is centered in the leveling sight (Figure 4). (The base of the pendulum will appear as a dark circle surrounded by a ring of light). Alternately, you may center the circle below the pendulum bob arm (see Figure 6B) in the circular hole.

Orient the Gravitational Torsion Balance so the mirror on the pendulum bob faces a screen or wall that is at least 1.5 meters away.
Figure 7: Using the Leveling Sight Figure 8: Adjusting the Height of the Pendulum
Vertical Adjustment of the Pendulum (generally only required the first time the Cavendish balance is used.)
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:

Grasp the torsion ribbon head and loosen the Phillips retaining screw (Figure 8a).

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 8b).

Tighten the retaining (Phillips head) screw.
Appendix III: Setting Up the Laser and the Screen

Replace the plastic cover with the aluminum cover.

Set up the laser about 50 cm from the Cavendish balance so it will reflect from the mirror to the screen where you will take your measurements (see Figures 9 & 10). Note that the Double Rod Clamp (to hold the polarizer) must be added to the rod now (as shown in Figure 10)! You will need to point the laser so that it is tilted upward toward the mirror and so the reflected beam clears the top of the laser to strike the screen. A white screen may be used, but the video is better using a brown screen (the back of a yellow note pad.)

In Figure 9, the spot that shows on the laser head above the laser output is due to reflection from the glass before the beam reaches the mirror. Initially adjust the laser head so that this beam passes just above the laser head and strikes the screen.

To get laser beam perpendicular to the Cavendish box, measure the distance of the vertical tube in the balance to a line perpendicular to the wall where the screen is. This might be the edge of the laboratory table or as in Figure 10, a convenient wall. Adjust the rod holding the laser head so that the laser beam (before it strikes the mirror) is at the same distance and parallel to the reference wall or table edge. Adjust the Cavendish box angle so the spot on the screen reflected from the glass (it isn’t moving) is at the same distance.

Adjust the tilt of the laser head so that the beam reflected from the mirror passes just above the laser head to strike the screen. The spot reflected from the glass should be on the laser head directly above the laser output as in Figure 9 (might even be below the laser head or strike the laser head).

Mount a transparent metric scale on the screen so that the spot reflected from the moving mirror traces back and forth directly above the transparent scale. The 15 cm mark on the scale should be at the “center” position measured in step 4.

Add the polarizer as shown in Figure 11. This is needed to control the intensity to prevent blooming with the video camera. First attach the Sensor Handle to the Polarizer using the plastic thumb nut. Then attach the Sensor Handle to the Double Rod Clamp and adjust so the beam passes through but low enough to not block the reflected beam. Note that only one polarizer is required since the laser beam is polarized.

Attach a copper wire to the grounding screw (Figure 12), and ground it to the earth.

Place the tungsten ball adaptors (aluminum rings) on the support stand (see Figure 11). These are necessary since the 1.5 kg lead balls originally used with this apparatus were somewhat larger and the large tungsten balls would be below the small balls without the adapter. Place the large tungsten masses on the support arm, and rotate the arm so the ball arm is roughly perpendicular to the case and the balls are as far from the case as possible.
Figure 9: Laser Setup Figure 10: Making Cav. Box Parallel to Screen
Figure 11: Polarizer Figure 12: Attaching the ground wire
Appendix IV. Rotational Alignment of the Pendulum Bob Arms  Zeroing (Generally only required the first time the Cavendish balance is used. May take many hours to do.) Do Appendices IIII first.
The pendulum bob arms must be centered rotationally in the case — that is, equidistant from each side of the case (Figure 13). Figures 14a and 14b illustrate what the goal is. The lower spot is the reflection off the glass. It does not move. The middle spot is the reflection off the mirror. The upper spot is reflected off the mirror, reflected back off the glass, and reflected a second time off the mirror. If the balls are centered in the case (assuming the mirror is parallel to the rod connecting the two small masses), then the three spots should be aligned vertically (as in Figure 14a) when the system is at rest. This is difficult to achieve. A more realistic case is shown in Figure 14b where the center spot is 1.5 cm to the right. The screen is at 1.5 m so the system is off by only 1/100 radian (0.50). This introduces a negligible error. To get these images, the video camera and polarizer have been removed.
Figure 13: Centering the Pendulum Bob Figure 14a: Figure 14b:
Perfect Alignment Realistic Alignment
Alignment procedure

Lower the screws holding the pendulum to release the pendulum. If the large balls are installed, put them at the center position. The pendulum will probably be striking the case on both sides (you will see the sudden rebound). If not, tap the vertical tube so that they do. Mark the position of the laser spot at each turning point. The positions may change by several millimeters due to nontorsional motions of the pendulum. This is not important. Mark the position halfway between the two turning points. This is where we would like the spot to be at equilibrium. Ideally, it would be directly above the fixed spot (reflection off of the glass) as in Figure 14a. However, the mirror is typically not perfectly aligned with the axis of the small balls, so when the balls are centered in the case, the laser spot is not quite on the line perpendicular to the axis. For the system used, Figure 14b actually shows the correct position of the laser spot at equilibrium.

Wait several hours until the pendulum comes to equilibrium and the laser spot is not moving. If the equilibrium position is against the case, this can generally be done much quicker since the period of oscillation is much shorter than eight minutes if the balls strike the case. If you observe that the spot strikes the case on one side but has a normal turning point before reaching the case on the other side, then equilibrium will either be against the case or close to it.

If the equilibrium position of the laser spot is within a few centimeters of the center position determined in step 1, you’re done! If the equilibrium position of the laser spot is not close to the center position determined in step 1, loosen the zero adjust thumbscrew (see Figure 15). It helps to put a pencil mark on the Zero Adjust Knob to better judge how far you turn it. Loosening the zero adjust thumbscrew will generally introduce large oscillations, but turning the zero adjust knob will not if you are gentle. The Zero Adjust Knob has a gear ratio of about 13:1 so a 90^{0} rotation will rotate the small ball’s equilibrium by about 7^{0}. This would result in a motion of the laser spot by about 14^{0}. The maximum the spot can move without hitting the case is about 20^{0} (11^{0} if using lead balls since they are larger.) If the equilibrium position is not against the case, skip to step 4c below. If the balls are against the case, turn the Zero Adjust Knob 90^{0} in the direction that will tend to move the balls off the case.

Observe the spot after an hour or so.

If it is still against the case on the same side, do another 90^{0} rotation with the zero adjust knob. Repeat as many times as necessary.

If it is against the case on the other side, do a 45^{0} rotation back in the opposite direction from before. That should result in an equilibrium position that does not touch the case.

If the equilibrium position is not against the case, wait about 4 hours until the spot comes to rest (+/ a few of millimeters). Measure the distance from the spot’s equilibrium position and the halfway point found in step 1 above. Calculate the angle represented by this distance (angle in radians = distance from halfway/distance of balance from screen). Divide this angle by two and rotate the zero adjust knob by this halfangle in the correct direction. Lock the Zero Adjust Thumbscrew. Wait four hours until the spot comes to rest. If it is not at the halfway mark, you may release the Zero Adjust Thumbscrew and repeat this step, but it won’t matter if the equilibrium position is off by several centimeters (see Appendix V)
Zero Adjust Knob
Zero Adjust Thumbscrew
Figure 15: Refining the Rotational Alignment of the Pendulum Bob
Appendix V: Small Corrections
Since we have claimed an uncertainty of only a couple of percent, we must consider small effects that might become important at this level of precision. We have identified seven effects that could be significant:

The laser is aimed slightly upwards so the beam goes over the laser head to hit the screen.

The mirror is not exactly vertical.

Mirror horizontal position is off due to not achieving perfect initial alignment and/or the mirror not being perfectly aligned with the axis of the small ball dumbbell.

Unless the initial alignment is perfect, the small balls are not initially centered in the case.

The angle between M and m tilts slightly as m moves.

The mass of the aluminum plates on the side of the case attracts the small balls.

The motion is not truly simple harmonic.
These effects are all small and partially canceling so the net effect is less than 0.1% and can be safely ignored. For those who are interested, details of the calculations are included in the remaining indices. We have ignored any interaction between the various corrections (for example, we calculated the force on the aluminum rod holding the small balls assuming it was on the symmetry plane of the case. Such small corrections to small corrections are surely ignorable.
1. The laser is aimed slightly upwards so the beam goes over the laser head to hit the screen. The laser must be tipped up by about 2^{0} for the reflected beam to clear the laser head at 50 cm.
Doesn't matter. Horizontal deviation, θ, depends only on the rotation of the mirror around a vertical axis. Only difference is that it means that the perpendicular distance to the screen is L (which is what was called for in the Procedure) and not the distance to the line where the spot moves. Really doesn't matter much since cos (tip angle) = cos 2⁰ = 0.9994. Correction = 0.00% (or +0.06% if measured distance to spot rather than perpendicular distance)
2. The mirror is not exactly vertical. This angle can be determined by comparing the reflected angle measured from horizontal to the incoming angle measured from horizontal. For the two PASCO Cavendish balances I checked, the mirror was tipped forward by 1.4^{0} and 2.3^{0}. I take 2^{0} as an average figure.
Angular velocity is a vector so the rotation speed about the tipped axis, ω', is given by
ω' = ω cos 2⁰ = 0.9994 ω → Correction = +0.06 %
3. Mirror horizontal position is off due to not achieving perfect initial alignment and/or the mirror not being perfectly aligned with the axis of the small ball dumbbell. The mirror error can be measured by noting where the small masses hit the case (see Appendix IV, step 1). For the two units I checked, the angles were 0.7^{0} and 0.9^{0}.
Figure 16: Angles
Here the mirror is rotated by θ₀ from the ideal position shown in black to the positions shown in blue. The laser beam comes in along the red vertical line and goes out along the red line. When the mirror rotates to the right the normal shifts by θ to the dotted green line and the laser to the solid green line. When the mirror rotates to the left the normal shifts θ (same torque) to the left to the dotted purple line and the laser to the solid purple line.
θ = θ_{2} + θ_{0} & θ = θ_{1} – θ_{0} → 2θ = θ_{2} + θ_{1}
S_{1} + S_{2} = (L tan 2θ_{1}  L tan 2θ_{0}) + (L tan 2θ_{2} + L tan 2θ_{0}) = L tan 2θ_{1} + L tan 2θ_{2}
This is the same as before provided we make the small angle approximation tan 2α = 2α. The maximum angle (hits case) is about 5^{0} for which tan is 0.25% larger than . More realistically, assuming a mirror angle error of 1^{0}, an alignment error of 0.75^{0} (2 cm at 150 cm) in the same direction plus the 0.67^{0} (the shift from equilibrium when the large masses are shifted), the angles should be less than 2.5^{0} and the using the small angle approximation makes our value for G about 0.06% low. Actually only about half this since if one of the above angles is 2.5^{0}, the other is only 1.2^{0} (since shifts in the opposite direction). Correction < +0.03%
4. Unless the initial alignment is perfect, the small balls are not initially centered in the case. We assume that the small mass dumbbell is 0.38^{0} off of the center of the case (laser spot off 2 cm at 150 cm).
Figure 17: Motion of m
Figure 17 shows the ideal case where the small masses are initially centered in the case and then
Δb = (d/2) sin θ = (10.1 cm/2) sin 0.350^{0} = 0.0308 cm
If the initial angle is off by 0.76^{0 }(laser spot off 2 cm at 10 cm), the masses are displaced by b_{0} where
b_{0} = (d/2)sin 0.76^{0} = 0.033 cm,
and the m to M distance becomes (b – [b + b_{0}]) for one position of the large masses and (b – [b – b_{0}]) for the other position. The forces in the two positions then become:
F_{1} = GmM/(b[Δb+b_{0}])² F_{2} = GmM/(b[Δbb_{0}])²
_{1} = 2F_{1}d = kθ_{1} τ_{2 }= 2F_{2}d = kθ_{2}
2d(F_{1}+F_{2}) = k(θ_{1}+θ_{2}) = k(2θ)
Thus the angle that the small masses swing through is directly proportional to the sum of F_{1}+F_{2}. Let F be the force when b_{0} is zero (perfect alignment), then
(F_{1}+F_{2})/(F+F) = [(b[Δb])² + (b[Δb])²]/[(b[Δb+b_{0}])² + (b[Δbb_{0}])²]
= [2(1[Δb]/b)²]/[(1[Δb+b_{0}]/b)² + (1[Δbb_{0}]/b)²]
(Δb = 0.0308 cm, b = 4.27 cm)
= [2(10.0308/4.27)²]/[(1[0.031+0.033]/4.27)² + (1[0.0310.033]/4.27)²]
= 1.0012
So the G value is too large by 0.12%. Correction due to b_{0} < 0.1
5. The angle between M and m tilts slightly as m moves.
Refer to Figure 17.
Δb = (d/2) sin θ = 0.0308 cm
b₁ = b  Δb = 4.27(1) cm  0.03 cm = 4.24(2) cm
c = d  d cos θ = 9.4 x 10^{5} cm
new distance = (b₁² + c²)^{1/2} = b₁ (we already assumed that)
α = tan^{1}(c/b₁) = 1.3x10^{3} degrees
perpendicular component of F = F cos = F cos 0.0013^{0}= F, so we can ignore the change in angle. Correction = 0
6. The mass of the aluminum plates on the side of the case attracts the small balls.
Assume the small mass m moves from halfway between the plates to a point b to the right. The right hand plate will then exert a force F_{+} which is slightly greater than the force exerted by the left hand plate, F_{}. We start by treating the plates as circles of radius R. df is the differential force exerted on m by the mass element dm. dF is the differential force exerted on m by the mass, dM, of the ring of radius r and lies along the axis of the plates. The plates are separated by 2w.
df cos = (Gm dm /[r^{2} + (wb)^{2}])([bb]/ [r^{2} + (wb)^{2}]^{1/2})
dF = df cos = Gm dm (wb)/[r^{2} + (wb)^{2}]^{3/2}) where the integral is evaluated around the ring of radius r.
dF = Gm dM (wb)/[r^{2} + (wb)^{2}]^{3/2}) = Gm(2rdr)(wb)/[r^{2} + (wb)^{2}]^{3/2})
where is the mass/area of the aluminum plate. = (2.7 g/cm^{3})(0.20 cm) = 0.54 g/cm^{2}
F_{+} = dF = 2Gm(wb)rdr/[r^{2} + (wb)^{2}]^{3/2} from r=0 to r=R.
F_{+} = 2Gm(1  (wb)/[r^{2} + (wb)^{2}]^{1/2})
Similiarly
F_{} = 2Gm(1  (w+b)/[r^{2} + (w+b)^{2}]^{1/2})
F_{net} /F = (F_{+ } F_{})/F = 2(b^{2}/M)((w+b)/[r^{2} + (w+b)^{2}]^{1/2}  (wb)/[r^{2} + (wb)^{2}]^{1/2})
where F is the force between m and M. The maximum circle that could be cut from the plate (centered on m) would have R = 2.5 cm. For this value of R
F_{net} /F = 0.0007
This is actually an upper bound since the function F_{net} /F is maximum for R = 2.1 cm. This may seem counterintuitive, but consider if the plates were really large. Then the force would be independent of the distance from the plate. Correction < 0.07%.
The motion is not truly simple harmonic.
It is clear from the goodness of fit of the damped sine wave to the data that the motion is very nearly simple harmonic, except that it is damped. This changes the frequency slightly. For the damped harmonic case:
ω = [ω₀²  B²]^{1/2 }= [(0.01289/s)²  (~3x10^{4}/s)²]1/2 = 0.0128865/s
Correction = 0
Written by Chuck Hunt
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