The Cavendish Gravitation Project

A DIY torsional pendulum measures the Universal Gravitational Constant G within a few % of the accepted value.

Newton’s Law of Gravity

Based on astronomical and laboratory measurements, Isacc Newton reasoned that the force of attraction (F) between two masses is proportional to the product of the masses $m_1$, $m_2$ and inversely to square of the distance r between the masses:

$F = \frac{G m_1 m_2}{r^2}$

The proportionality constant G is called the Universal Gravitational Constant. It took 111 years after Newton’s publication of this equation to obtain a measurement of G.

Using units of kilograms for m, meters for r, and $kg m/s^2$ (Newtons) for F, $G = 6.67 \times 10^{-11} m^3/kg s^2$.

Examples

The equation $F = G \frac{m_1 m_2}{r^2}$ describes the orbits and trajectories of baseballs, moons, planets, galaxies, and the weight of objects on Earth.

The weight of an object on Earth is a measure of the gravitational force between the object and the Earth. If we consider only the objects on the surface then using $r = R_e$, the radius of the Earth, $m_2 = M_e$, the mass of the Earth, $m_1 = m$, the mass of the object, Newton's gravitation equation can be re-written as

$F = G \frac{m M_e}{{R_e}^2} = m g$

$g = G \frac{M_e}{{R_e}^2} $

The quantity g is Earth’s gravitational acceleration. It is the same acceleration for any object near the Earth’s surface. Every second that something falls it gains about 9.8 m/s more velocity.

The Earth’s radius (Re) was measured and calculated about 240 BC by Eratosthenes. The acceleration g can be readily measured by tabulating the distance and time of fall for an object. If we also know the universal G value, then the mass of the Earth could be calculated using:

$M_e = \frac{g}{G}{R_e}^2$

Measuring G:

Henry Cavendish (1798) used a torsional balance made 45 years earlier by John Michell (1753). The instrument and Cavendish’s meticulous efforts generated a value of G with an accuracy unsurpassed for almost 100 years.

A torsional balance (pendulum) was formed by connecting two spheres of mass $m_1$ by a rod of length L, suspended in the center by a fine wire. Larger masses $m_2$ were brought close to the $m_1$ spheres to gravitationally attract them, which twisted the wire by a tiny angle $\theta$. Knowing the wire’s twist properties, $\theta$, the masses, the distance r between the centers of the spheres, and L, you can calculate G.

The Cavendish Apparatus:

Large lead spheres (12” diameter, 158 kg or 348 lbs) were placed in various positions to attract the smaller lead spheres (2” diameter, 0.78 kg or 1.7 lbs). A “silvered copper wire” was used for suspension. The distance r between the centers of the spheres was r = 0.22 m. Knowing the value of G today, the gravitational force can be calculated to be $F = 1.7 \times 10^{-7}$ N (38 nano lbs). The displacement of the $m_1$ spheres when switching between position 1 and 2 was about 0.1” and was measured with high accuracy using telescopes protruding through the walls. The suspension wire had a torsional spring constant (discussed later) of $K_{wire} = 6.35 \times 10^{-5}$ mN/rad, leading to an oscillation period of T = 15 minutes.

This was a massive apparatus. The project goal is to create a much smaller torsional pendulum with at least 5% accuracy at relatively low cost.

Key physics of a gravitational torsion balance:

Two large masses $m_2$ have their centers each a distance r from the centers of smaller masses $m_1$. The gravitational force F between each $m_1-m_2$ pair provides a torque on the pendulum and twists the suspension wire. The $m_2$ mass centers are joined by a thin rod of length $L_{rod}$. The total gravitational torque is balanced by the twisting torque from the wire (with torsional spring constant K), resulting in a twist angle $\theta$. In equation form (approximately):

$\tau_{wire} = K \theta = \tau_G = 2(L_{rod}/2) F$

$F = \frac{G m_1 m_2}{r^2}$

Combining these equations, the angle of rotation of the wire due to gravitational attraction is then

$\theta = \frac{G L_{rod} m_1 m_2}{K r^2}$

To determine G, we need to measure $\theta, L_{rod}, r, m_1, m_2$, and K.

Project Design:

To design the apparatus, we need to choose values for $m_1, m_2, L_{rod}, r$ and K that maximize the rotation angle $\theta$. To achieve high values of $\theta$, we need the masses and $L_{rod}$ to be large, with K and r small. From a practical standpoint, $m_1$ needs to be small enough for the wire to support the pendulum without breaking. The smallest r occurs when $m_1$ and $m_2$ are in contact and r is the sum of their radii, $r = R_1 + R_2$. The mass is the density $\rho$ times the volume. Then for $R_2 \gg R_1$:

$\theta \approx (\frac{4\pi}{3})^2 G L_{rod} \rho^2 R_2 \frac{{R_1}^3}{K}$

This tells us that the density and radius of the large mass should be maximized. This is why Cavendish used a sphere of lead 1 foot in diameter.

A chart of densities shows that lead and tungsten are good choices. Tungsten is rare and rather expensive. Moderate density metals include brass, copper and iron. Iron invites the possibility of magnetic forces, which could overwhelm the gravitational force. The ease of machining brass compared to copper favors brass as a logical metal for a home shop experiment. Lead can also be used via a molding process described later.

Gravitational force between various shapes vs. Sphere + Sphere

Cavendish used spheres for $m_1$ and $m_2$. The difficulty of creating high quality large metal spheres in a home shop prompted a consideration of other shapes, like blocks and cylinders. How does this change the physics?

The basic form of Newton’s Law of Gravitation is:

$F = G \frac{m_1 m_2}{r^2}$

where r is measured between the centers of each sphere. Does this equation still apply to non-sphere shapes? If the $m_2$ mass was a tall rod instead of a sphere, its total mass would increase linearly with the rod length, but a $\frac{1}{r^2}$ dependence of the force between $m_1$ and the tip of $m_2$ would rapidly weaken with distance, unlike for a spherical $m_2$.

To explore this effect, shapes can be divided into small elements (voxels) of volume dV with masses $dm_1$ and $dm_2$, separated by a distance $r_{12}$. Each $dm = \rho dV$, where $\rho$ is the object density, giving the force between the voxels dF:

$dF = G \frac{\rho_1 \rho_2 dV_1 dV_2}{{r_{12}}^2}$

The gravitational force dF between these small masses can be added up (as vectors) to obtain the total force. As an approximation, the voxels can be tiny cubes within the boundary of the shape. This approach makes the computation relatively simple and fast for a computer.

As a check on the voxel calculation method, the case for two spheres is shown below, where the diameter of the larger mass is varied relative to the smaller one (width2/width1) and the separation ysep along y, relative to width1 are varied. The results confirm Fy is equal to $F0 = G \frac{m_1 m_2}{{ysep}^2}$ for all the variations.

Various shape combinations were examined with height = width for each object. The gravitational force Fy between $m_1$ and $m_2$ along y was calculated as a function of separation of the centers, ysep (normalized to width1 of $m_1$ ), and compared to the result for spheres $F0 = G \frac{m_1 m_2}{{ysep}^2}$ ($\frac{Fy}{F0} = 1$ for spheres).

The results show that the gravitational force between two masses is not exactly modeled by $F0 = G \frac{m_1 m_2}{{ysep}^2}$, except for the case of two spheres, or if the separation becomes significantly larger than the shapes. For example, for the case of a 1” cube of mass $m_1$ separated from a cylinder width2 = 3” (diameter and height) of mass $m_2$, the force can be as low as 87% of F0 if the objects are in contact. We can refer to this as the Fy/F0 factor.

Other Shape Considerations

Another aspect to consider for various shapes is the mass relative to that of a sphere of the same width. For a given material, a cube of width w has a larger mass than a sphere of the same width by $\frac{w^3}{\frac{4\pi}{3}({\frac{w}{2}})^3} = \frac{6}{\pi} = 1.91$. A cylinder of diameter and height w will have a mass 1.5 that of a sphere of the same diameter. A comparison of the gravitational force Fy between two shapes (equal density), relative to that of two spheres Fyss (including the shape effect discussed above), for the case of width1 = 1”, width2 = 3”, and ysep = ymin + 0.5” (= 2.5”) is shown in the chart below. The differences are mostly the result of the volume fill of each shape and indicate that shapes other than spheres can significantly improve the sensitivity of the measurement for given widths and density of materials.

Additionally, the requirement of symmetrically rotating the large $m_2$ masses about z, dictates that they should also be symmetric about z to avoid having to rotate them and adding unwanted vibration. This limits the shape of $m_2$ to cylz or sphere. The optimal choice then appears to be block ($m_1$) – cylinder along z ($m_2$).

The Masses

Based on the these arguments, the $m_1$ mass was chosen to be a 1” cube, giving $m_1$ = 137g. These were drilled halfway through for a ¼” brass rod that was soldered to the cubes, making $L_{rod}$ = 0.229 m (center to center on cubes).

Despite the potential hazards of working with lead, a method was devised to create a pendulum consisting of two 1" cubes of high purity lead connected by a 1/4" aluminum rod (same dimensions as the brass pendulum).

The lead cubes were made as follows:

The mold was made from a section of aluminum channel. A ¼-20 bolt was used to mold a threaded hole in the lead cube to screw onto the ends of a threaded aluminum rod.
Aluminum foil was used as a gasket to seal the bottom cap of the mold.
High purity lead was melted (outdoors with respirator) and poured into the mold, cooled and pushed out. Lead must first be completely removed from the inner surfaces to facilitate release from the mold.

The $m_2$ mass was chosen for similar reasons to be a solid cylinder with height and diameter of 3”, machined by hobby machinist Keith on a larger lathe, but it is possible on a mini-lathe. This results in each $m_2$ = 2.99 kg (6.59 lbs). A ¼” diameter pin was added to align the cylinders into placement holes.

Total Torque

In the experiment configuration, there are two large masses $m_2$ and a tortional pendulum with two masses $m_1$. The main force acting on the $m_1$ masses is from the $m_2$ mass nearest them (a distance r), $F = \frac{G m_1 m_2}{r^2}$. There is also an attraction between each $m_1$ and the more distant $m_2$ mass that causes a counter torque, but it is smaller because the separation is larger. The distance between the farthest masses $r’ = \sqrt{{L_{rod}}^2 + r^2}$, leading to a force $F’ = G \frac{m_1 m_2}{r’^2}$. The total torque between the nearest masses for rotation about the pendulum center is $\tau = -2 \frac{L_{rod}}{2} F$, and between the more distant masses is $\tau’ = +2 \frac{L_{rod}}{2}\frac{r}{r’} F’$ (the $\frac{r}{r’}$ factor is the sine of the angle between F’ and the rod). The total torque on the suspension wire is then:

$\tau + \tau’ = -L_{rod} G \frac{m_1 m_2 (1 – \frac{r^3}{r’^3})} {r^2}$

We can refer to $B = 1 – \frac{r^3}{r’^3}$ as the “torque factor”. In this experiment, $r \approx 2.5”$, $L_{rod} = 9”$, so r’ = 9.34”. The torque factor B is 0.98, meaning a 2% reduction to the torque due to the opposite $m_2$ mass.

Measuring Wire K

As a torsional pendulum rotates, the wire twists and resists the rotation, causing the pendulum to oscillate back and forth, repeating every T seconds. The $m_1$ masses, $L_{rod}$ and the period T are related to the wire torsional spring constant K by: $I (\frac{2\pi}{T})^2 = K$, where I is the moment of inertia of the barbell $I = 2 m_1 (\frac{L_{rod}}{2})^2$. This gives:

$K = 2 m_1 (\frac{L_{rod}}{2})^2 (\frac{2\pi}{T})^2 = 2 m_1 (\frac{\pi L_{rod}}{T})^2$

K can therefore be determined by measuring $m_1, L_{rod}$, and the oscillation period T.

Optimizing Wire K

The suspension wire is crucial to this experiment. Since the gravitational force is very weak, K must be small to allow a significant twist of the wire so that $\theta$ can be large enough to measure. For Cavendish’s experiment, the barbell oscillation period T was about 15 minutes, $L_{rod}$ = 186 cm, $m_1$ = 778g. The oscillation equation $K = 2 m_1 (\frac{\pi L_{rod}}{T})^2$ then tells us that $K = 6.56 \times 10^{-5}$ m N/ rad.

To see what this means for materials and diameter of the wire, the value of K can be calculated from (https://en.wikipedia.org/wiki/Torsion_constant):

$K = \frac{\pi}{2} G’ \frac{r_{wire}^4}{L_{wire}}$

G’ is the modulus of rigidity that depends on the material. From the Cavendish paper, $L_{wire}$ = 102cm, and from the ratio mass/length of the wire implies $r_{wire} \approx 0.13mm$. Using the above value for K gives $G’ \approx 1 \times 10^{11}$ Pa, on the order of the value for steel.

The K equations tell us that T should be proportional to $\frac{1}{r_{wire}^2}$. To verify this, T was measured for a series of wire radii ($r_{wire}$ = 0.05 mm to 0.2mm) made from stainless steel SAE 316L (music wire). The plot (below) of T vs. $\frac{1}{r_{wire}^2}$ is linear and its slope gives the expected wire G’ value $\approx 10^{11}$ Pa.

The wire must also suspend a weight $w = 2 m_1 g$ without stretching or breaking. This depends on the wire’s cross sectional area $A = \pi r_{wire}^2$ and its yield strength $U = \frac{w}{A}$. While smaller $r_{wire}$ reduces K to increase $\theta$, it also reduces the maximum $m_1$, which lowers the gravitational force and $\theta$. These relations give the maximum $m_1 = U \frac{\pi}{2g} r_{wire}^2$, so we can write $\theta$ as:

$\theta = (G L_{rod} \frac{m_2}{r^2}) (\frac{m_1}{K}) = (G L_{rod} \frac{m_2}{g r^2}) (U \frac{L_{wire}}{G’ r_{wire}^2})$

This shows that the wire must have a high yield strength U, low modulus of rigidity G’, small radius, and long length.

A series of stainless-steel SAE 316L wires were tested by suspending weights to determine U values using the mass at the breaking point $m_{max} = U \frac{\pi}{g} r_{wire}^2$. The results are plotted below and give a value for $U = 6.2 \times 10^8$ Pa, consistent with the accepted value for SS. There was considerable stretching near the maximum mass value for this material.

For the chosen $m_1$ masses and connecting rod, the total barbell mass = 337g. The stress measurements indicate that the wire radius for SAE 316L should be no smaller than 0.05mm ($\approx$ 38 AWG) to minimize stretching. Using $L_{wire}$ = 0.318m (for this apparatus) gives $K \approx 2 \times 10^{-6}$ m N/ rad ($\approx \frac{1}{30}$ that used by Cavendish). Unfortunately, the $\theta$ value calculated from this (using the accepted value for $G = 6.67 \times 10^{-11} m^3/kg sec^2$) is only about 0.05 degrees, which is very difficult to measure.

After searching materials tables for values of G’ or U, tungsten stands out as a good choice. Its value for U = 1510 MPa, over twice that of the stainless-steel wire. It is more expensive (about $150/ 25’ for 44 gage), but tests showed that the barbell mass could be suspended without stretching or breaking at

$r_ {wire}$ = 0.025mm (44 gage), increasing the expected $\theta$ by $4\times$. The measured oscillation period is then about 11 minutes, giving $K = 3.2 \times 10^{-7}$ m N/ rad.

Conclusion: Use 44 gage tungsten wire.

Calculating the Moment of Inertia

The moment of inertia of the pendulum (required to calculate the wire K from the oscillation period T) is $I_0 = 2 m_1 (\frac{L_{rod}}{2})^2$. This comes from the definition, summing all the mass elements times their distance from the pivot squared. A more exact sum of all the points in the shape plus the connecting rod (radius $r_{rod}$) gives the following results for shapes of the same width w (girth = height):

$I_{spheres} = I_0 + \frac{1}{12} m_{rod} (L_{rod} – w)^2 + 2(\frac{1}{10} m_1 w^2)$

$I_{cubes} = I_0 + \frac{1}{12} m_{rod} (L_{rod} – w)^2 + 2(\frac{1}{6} m_1 w^2)$

$I_{cylz} = I_0 + \frac{1}{12} m_{rod} (L_{rod} – w)^2 + 2(\frac{1}{8} m_1 w^2)$

The last terms are unique to the shapes. The center terms are the moment of inertia of the rods ($I_{rod}$).

In this experiment, $L_{rod}$ = 9”, w = 1”, $m_{rod}$ = 55.3g (brass), $m_1$ = 141g (brass cube), giving: $I_0 = 3.9 \times 10^{-3} kg m^2 , I_{rod} = 1.9 \times 10^{-4} kg m^2$ (5% addition) and the last term = $1.5 \times 10^{-5} kg m^2$ (0.4% addition).

In conclusion, the sum of $I_0$ and the rod moment of inertia is sufficient.

How to Measure $\theta$

Cavendish used a vernier scale at both ends of the pendulum to measure the deflection of the $m_1$ masses. In modern experiments, a laser beam incident on a mirror attached to the pendulum reflects at an angle $2\theta$, producing a spot at position x on a screen a distance D from the apparatus. As the figure indicates, $\theta = \frac{1}{2} \arctan{\frac{x}{D}} \approx \frac{x}{2D}$ for $D \gg x$.

For example, if the pendulum is deflected $\theta = 0.1^\circ$, and D = 8.6 meters, the laser spot will deflect x = 3cm on the screen. If the accuracy limit of x is 2mm due to the beam size, then the detection limit for $\theta$ would be $\frac{0.002}{2 \times 8.6}$ radians or $0.0067^\circ$.

In the measurements, the $m_2$ masses are moved from position 1 to position 2 as shown below, the pendulum masses will respond by rotating from $+\theta$ to $–\theta$, and the laser spot will move from +x to –x, where $x = 2D\theta$. Measuring the difference $\Delta x$ between the laser spots for these two positions gives $\theta = \frac{\Delta x}{4D}$.

Physics Summary

Torque balance: $K\theta = \tau$

Total torque: $\tau = L_{rod}F B$

Gravitational force: $F = \frac{G m_1 m_2}{r^2}\frac{Fy}{F0}$

Tortional spring constant: $K = I (\frac{2\pi}{T})^2$

Moment of Inertia: $I = 2 m_1 (\frac{L_{rod}}{2})^2 + m_{rod} \frac{(L_{rod} – w)^2}{12}$

Theta measurement: $\theta = \frac{\Delta x}{4D}$

Final Result:

$G = \frac{\frac{\Delta x}{D}\frac{1}{T^2} \pi^2 \frac{L_{rod}}{2} r^2}{m_2 B \frac{Fy}{F0}}$

The Apparatus

Note: Some of the concepts used in this apparatus were inspired by images of a system sold by PASCO company, makers of outstanding educational experiments. Many of the features came from the advice and experience of Hobby Machinist Keith.

Design Considerations

The goal is to measure very slight changes in the angle of a torsional pendulum due to gravitational attraction.
Competing forces include air currents, static electricity, friction and magnetic effects.

Air currents were minimized by enclosing the pendulum in a box, including a sealed window.
The metal housing was connected to ground to minimize static effects.
Aluminum and brass hardware were used to reduce magnetic forces.

The pendulum was centered to avoid any contact with the housing.
The initial twist in the wire was substantially reduced using an adjustment knob at the top of the wire.
A clamp was designed to minimize kinking of the wire to prevent wire breakage.
A clamp for the pendulum was included to relieve strain on the wire when not in use.
The mirror is attached to the pendulum clamp and is of optical quality to achieve a small reflected spot at the screen.

Front cover removed:

Top and Bottom Wire Clamps

The top clamp consists of a brass rod with a small hole for the suspension wire, topped with a knurled knob, plus a bronze bushing, and a clamp piece for the wire. The wire is fed through an aluminum sleeve and the center tube. The sleeve allows the top clamp to slide up and down the center tube for height adjustment.

The bottom clamp is centered on the pendulum rod, then the lower screw is tightened. The wire is fed through the top hole, the top portion is screwed in place over the wire and the clamp screw is tightened. Once suspended, the base plate levels are adjusted to center the plunger within the bottom cavity.

Height Adjustment of Pendulum and Top of Wire

Top Views of Pendulum and Large Masses

Measurement Procedure

With the laser aimed at the center of the pendulum mirror, orient the apparatus and laser to get the reflected beam onto the screen a distance D from the apparatus. In this case, D = 8.28 meters (27.17 feet).
With the large masses rotated away from the pendulum housing, carefully adjust the angle bias knob on the wire to get the reflected beam oscillating about the center of the screen. Since the oscillation period is more than 10 minutes, this requires long wait times between very fine adjustments.
Set up a screen (such as white poster board) with a tape measure adhered to the screen to measure x.
Set up a smart phone on a tripod to record time-lapse videos of the full screen.
Gently swivel the large masses into proximity of the enclosure, just touching the housing walls on either side.
Record the video for at least 4 complete oscillations (in this case, recordings were made for 90 minutes). Note the positions and time on the screen. Save the recording.
Gently swivel the large masses to the opposite position and record the oscillations as before.
Repeat with the masses in opposite positions several times.
Analyze the videos to determine the period of oscillation and the left and right equilibrium x values, x0, calibrated by a measuring tape on the screen (see the next section).

The Use of Time Lapse Videos

The period of oscillation T is approximately 11 minutes. Typically, 8 - 9 oscillations were observed for each position of the large masses, over 3 – 4 repeats for both mass positions, giving a total measurement time of 4.4 to 6.6 hrs. By using a time lapse video of the laser spot vs. time, the apparatus could be left on its own, changing the $m_2$ positions about every 45 minutes. A clock was included to record real time, as seen in the video below. The position x was measured by calibrating the image pixels to the tape measure, and time was derived by the frame count, calibrated to the clock. The extremes in x and their time were recorded to derive the oscillation period and the center point of each oscillation for $m_2$ in position 1 and position 2. The range of oscillation exponentially decreased over time due to torsional friction. Adjusting the $m_2$ positions always led to vertical oscillations from vibrational modes of the pendulum, which eventually diminish.

Time Lapsed Video of the laser spot oscillation:

Example data:

The maximum and minimum x positions of the laser spot were extracted from the time lapse videos along with the time for the $m_2$ masses in their two positions. The average x0 were calculated to find Delta x = x0(position 1) - x0(position 2).

Results and Calculations for Brass Pendulum:

The experimental values for Delta x and T and the other parameters were used in the previously derived equation to estimate G. The result is 95.6% of the accepted value.

Results and Calculations for Lead Pendulum:

The higher $m_1$ mass for the Lead pendulum gives a higher oscillation period T with the same suspension wire and a higher angular shift (larger Delta x) due to higher gravitational force. The result is 101% of the accepted value.

Overall, this apparatus and measurements are within a few % of the accepted value for G.

Mass of the Earth

The mass of Earth can be calculated from the radius of the earth $R_e = 6.36 \times 10^6$ m, the gravitational acceleration g, and the two values for G found in this measurement: $M_e = \frac{g R_e^2}{G} = 6.21 \times 10^{24} kg, 5.88 \times 10^{24} kg$, compared to the accepted value of $5.97 \times 10^{24} kg$.

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