The Cavendish experiment, conducted by English scientist Henry Cavendish between 1797 and 1798, holds immense historical significance as the first experiment performed in a laboratory setting to measure the gravitational force between masses. Although Cavendish originally expressed his findings in terms of the relative density or mass of the Earth, the experiment was also the first to produce accurate values for the gravitational constant, \(G\). Cavendish himself famously referred to his experiment in correspondence as “weighing the world”. The accurate values derived from this work for these geophysical constants were groundbreaking for the time.
What Was the Original Goal and Who Invented the Concept of the Cavendish Experiment?
The concept for the experiment was originally developed by English geologist John Michell sometime before 1783. Michell even constructed a torsion balance apparatus for this purpose. However, Michell died in 1793 before he could finish the work. Following Michell’s death, the apparatus was passed first to Francis John Hyde Wollaston, and then to Cavendish, who rebuilt the equipment while retaining the core of Michell’s original design. Cavendish fully credited Michell for devising the experiment. Cavendish carried out a series of measurements with the equipment and published his findings in the Philosophical Transactions of the Royal Society in 1798.
The formulation of Newtonian gravity using a gravitational constant (\(G\)) did not become standard until much later, with one of the first explicit references to \(G\) appearing in 1873, 75 years after Cavendish’s publication. Consequently, Cavendish’s primary objective was to determine the density of the Earth (\(\rho_{\text{earth}}\)), which, alongside the known radius of the Earth (\(R_{\text{earth}}\)), served the function of an inverse gravitational constant in the unit conventions of the era.
How Did Henry Cavendish Build the Torsion Balance Apparatus to Measure Gravity?
The foundation of the Cavendish apparatus was a torsion balance. This consisted of a six-foot (1.8 m) horizontal wooden rod suspended from the center by a fine torsion wire. At either end of this rod were two small, spherical lead weights. These small masses were 2 inches (51 mm) in diameter and weighed 1.61 pounds (0.73 kg) each.
The attractive force was provided by two massive lead balls. These large balls were 12 inches (300 mm) in diameter and weighed 348 pounds (158 kg) each. They were suspended separately and could be rotated by a pulley from outside the housing. The large masses could be positioned either away from or to either side of the small balls, placed 8.85 inches (225 mm) away.
The faint gravitational attraction between the small and large balls caused the horizontal rod to rotate slightly, twisting the suspension wire. The rod rotated until the twisting force (torque) exerted by the wire balanced the combined gravitational force of attraction between the pairs of masses.
Measurement Sensitivity and Environment
The force involved in twisting the balance was incredibly small, measuring only \(1.74 \times 10^{-7}\) Newtons. This force is equivalent to the weight of about 0.0177 milligrams, or approximately 1/50,000,000 of the weight of the small balls.
To achieve accurate measurements of such a minute force, Cavendish demonstrated remarkable sensitivity for the time:
- He housed the entire apparatus inside a mahogany box (approximately 1.98 meters wide, 1.27 meters tall, and 14 cm thick), which was itself placed within a closed shed on his estate.
- This elaborate housing was necessary to prevent air currents and temperature changes from interfering with the measurements.
- Cavendish observed the slight movement of the rod through two holes in the shed walls using telescopes.
- The key observable deflection of the rod was measured to be about 0.16 inches, though only 0.03 inches when a stiffer suspending wire was utilized.
- Cavendish achieved an accuracy better than 0.01 inches (0.25 mm) by using vernier scales at the ends of the rod.
To quantify the twisting properties of the wire—the torsion coefficient (\(\kappa\))—Cavendish timed the natural period of oscillation (\(T\)) of the balance rod as it moved clockwise and counterclockwise. For his initial three experiments, the period was roughly 15 minutes, but after replacing the wire with a stiffer one for the following 14 experiments, the period decreased to about 7.5 minutes. Since the rod was constantly oscillating, Cavendish had to measure the deflection angle while it was in motion. The accuracy of Cavendish’s result remained unmatched until C. V. Boys’ experiment in 1895. Today, variations of Michell’s torsion balance remain the dominant method for measuring the gravitational constant (\(G\)).
What Was Cavendish’s Original Result for Earth’s Density and How Accurate Was It Compared to Modern Values?
Cavendish determined the force between the pairs of masses by measuring the angle of the rod and knowing the wire’s torque. Since the gravitational force exerted by the Earth on the small ball could be found directly by weighing it, the ratio of the two forces allowed Cavendish to calculate the relative density of the Earth, using Newton’s law of gravitation.
Cavendish concluded that the Earth’s density was \(5.448 \pm 0.033\) times the density of water.
| Geophysical Constant | Cavendish’s Value | Modern Accepted Value | Difference (approx.) |
|---|---|---|---|
| Earth’s Density (\(\rho_{\text{earth}}\)) | \(5.448 \text{ g/cm}^3\) | \(5.514 \text{ g/cm}^3\) | ~1.2% |
| Gravitational Constant (\(G\)) | \(6.74 \times 10^{-11} \text{ m}^3 \text{ kg}^{-1} \text{ s}^{-2}\) | \(6.67408 \times 10^{-11} \text{ m}^3 \text{ kg}^{-1} \text{ s}^{-2}\) (CODATA 2014) | ~1.0% |
Note on Error: An arithmetic error was later discovered in 1821 by Francis Baily, meaning the erroneous value of \(5.480 \pm 0.038\) appeared in Cavendish’s original paper.
This result provided crucial evidence supporting the idea of a planetary core made of metal. Cavendish’s determined density (5.4 g/cm³) was 23% higher than the result from the earlier 1774 Schiehallion experiment. The result is close to 80% of the density of liquid iron and 80% higher than the density of the Earth’s outer crust, thereby strongly suggesting the existence of a dense iron core.
Though Cavendish did not determine \(G\) explicitly, later authors reformulated his density results into modern units. Converting Cavendish’s value for Earth’s density (5.448 g/cm³) into SI units yields a value for the gravitational constant: \(G = 6.74 \times 10^{-11} \text{ m}^3 \text{ kg}^{-1} \text{ s}^{-2}\). This value is remarkably accurate, differing by only 1% from the 2014 CODATA accepted value.
What Equations Do Modern Physicists Use to Calculate G and Earth’s Mass from the Cavendish Data?
Modern physicists utilize the data collected from the Cavendish experiment to calculate \(G\) and the Earth’s mass (\(M_{\text{earth}}\)), although the mathematical method differs from Cavendish’s original approach.
1. Finding the Force Balance (Equilibrium)
At equilibrium, the gravitational torque (\(LF\)) must equal the restorative torque of the wire (\(\kappa \theta\)):
\(\displaystyle \kappa \theta = LF\)Using Newton’s law of universal gravitation:
\(\displaystyle F = \frac{GmM}{r^{2}}\)This leads to:
\(\displaystyle \kappa \theta = L\frac{GmM}{r^{2}} \qquad \text{(Equation 1)}\)Where \(\kappa\) is the torsion coefficient, \(L\) is the length of the torsion beam, \(M\) and \(m\) are the large and small masses, and \(r\) is the distance between their centers.
2. Finding the Torsion Coefficient (\(\kappa\))
The value \(\kappa\) is determined by measuring the natural resonant oscillation period (\(T\)) of the torsion balance:
\(\displaystyle T = 2\pi \sqrt{\frac{I}{\kappa}}\)Assuming the mass of the torsion beam is negligible, the moment of inertia (\(I\)) is attributed solely to the small balls:
\(\displaystyle I = \frac{mL^{2}}{2}\)This gives:
\(\displaystyle T = 2\pi \sqrt{\frac{mL^{2}}{2\kappa}}\)3. Calculating G
Solving the period equation for \(\kappa\), substituting into Equation 1, and rearranging for \(G\) gives the final expression for the gravitational constant:
\(\displaystyle G = \frac{2\pi^{2}Lr^{2}\theta}{MT^{2}}\)4. Calculating Earth’s Mass and Density
Once \(G\) is known, the mass of the Earth (\(M_{\text{earth}}\)) can be calculated by equating the weight of an object (\(mg\)) at the Earth’s surface to the gravitational force exerted by the Earth:
\(\displaystyle M_{\text{earth}} = \frac{gR_{\text{earth}}^{2}}{G}\)Finally, the Earth’s density (\(\rho_{\text{earth}}\)) can be found using the mass and radius:
\(\displaystyle \rho_{\text{earth}} = \frac{3g}{4\pi R_{\text{earth}}G}\)Where \(g\) is the acceleration of gravity at the surface of the Earth, and \(R_{\text{earth}}\) is the radius of the Earth.
Can the Cavendish Experiment Be Successfully Demonstrated in a Classroom Setting?
Yes, the Cavendish experiment can be successfully reproduced to provide a qualitative visual demonstration that gravity exists between all objects, even though it is an incredibly weak force.
In one reproduction attempt, a torsion balance was created using two meter sticks taped together, suspended by a copper wire, with two one-kilogram masses held on either end. The apparatus was noted to be extremely sensitive to air currents, requiring the demonstration to be performed on a quiet weekend.
- For the massive attracting objects, lead bricks were used, weighing about 12.03 kg (roughly 25 to 30 pounds).
- Lead is a desirable material because it is highly dense and takes up relatively little volume. Using a small volume is important because gravitational effects depend heavily on the inverse square of the distance (\(r\)) between the objects, meaning that keeping the distance small maximizes the measurable force.
The modern demonstration established the “noise” of the experiment—the natural oscillation (amplitude) of the torsion balance before the massive bricks were introduced. Once the lead bricks were placed close to the small masses, the footage showed a clear effect: the torsion balance slowed dramatically and changed direction earlier than it would typically. This constant acceleration towards the lead bricks was attributed to the gravitational force exerted by the bricks on the hanging masses. While this specific reproduction was not designed as a quantitative measurement of \(G\), it confirmed that the weak gravitational attraction between laboratory-sized objects can indeed be visually detected.
