Henry Cavendish (1731-1810): Weighing the Earth
In 1686 Isaac Newton realized that the motion of the planets and the moon as well as that of a falling apple could be explained by his Law of Universal Gravitation, which states that any two objects attract each other with a force equal to the product of their masses divided by the square of their separation times a constant of proportionality.
Newton was not particularly concerned to evaluate the constant of proportionality, G, for two reasons. First, a consistent unit of mass was not in widespread use at the time. Second, he judged that since the gravitational attraction was so weak between any pair of objects whose mass he could sensibly measure, being so overwhelmed by the attraction each feels toward the center of the earth, any measurement of G was impractical. However, Newton estimated this constant of proportionality, called G, perhaps from the gravitational acceleration of the falling apple and an inspired guess for the average density of the Earth.
More than 100 years elapsed before G was first measured in the laboratory; in 1798 Cavendish and co-workers obtained a value accurate to about 1%. When asked why he was measuring G, Cavendish replied that he was "weighing the Earth"; once G is known the mass of the Earth can be obtained from the 9.8m/s2 gravitational acceleration on the Earth surface and the Sun's mass can be obtained from the size and period of the Earth orbit around the sun. Early in this century Albert Einstein developed his theory of gravity called General Relativity in which the gravitational attraction is explained as a result of the curvature of space-time. This curvature is proportional to G.
Naturally, the value of the fundamental constant G has interested physicists for over 300 years and, except for the speed of light, it has the longest history of measurements. Almost all measurements of G have used variations of the torsion balance technique pioneered by Cavendish. The usual torsion balance consists of a 'dumbbell' (two masses connected by a horizontal rod) suspended by a very thin fiber. When two heavy attracting bodies are placed on opposite sides of the dumbbell, the dumbbell twists by a very small amount. The attracting bodies are then moved to the other side of the dumbbell and the dumbbell twists in the opposite direction. The magnitude of these twists is used to find G.
Some history:
Spurred by his interest in the structure and composition of the interior of the earth, Henry Cavendish in a 1783 letter to his friend Rev. John Michell discussed the possibility of devising an experiment to "weigh the earth." Borrowing an idea from the French scientist Coulomb who had investigated the electrical force between small charged metal spheres, Michell suggested using a torsion balance to detect the tiny gravitational attraction between metal spheres and set about constructing an appropriate apparatus. He died in 1793, however, before conducting experiments with the apparatus.
The apparatus eventually made its way to Cavendish's home/laboratory, where he rebuilt most of it. His balance was constructed from a 6-foot wooden rod suspended by a metal fiber, with 2-inch-diameter lead spheres mounted on each end of the rod. These were attracted to 350-pound lead spheres brought close to the enclosure housing the rod. He began his experiments to "weigh the world" in 1797 at the age of 67, and published his result in 1798 that the average density of the earth is 5.48 times that of water. His work was done with such care that this value was not improved upon for over a century. The modern value for the mean density of the earth is 5.52 times the density of water. Cavendish's extraordinary attention to detail and to the quantification of the errors in this experiment has lead many to describe this experiment as the first modern physics experiment.
In order to measure G we need refined and delicate laboratory apparatus (torsion balance) and experimental design, in which a multitude of subtle effects must be compensated for or canceled out. We, however, aren't going to measure anything - we're only interested in observing universal gravitation. This allows simplifying the torsion balance to something we can set up in the basement.
The following inspiring experiment was suggested by John Walker, Fourmilab Switzerland. It is strongly recommended to read his full article in order to perform correctly this experiment:
Bending Spacetime in the Basement
An underground room is ideal because it minimises temperature variations and vibration which might perturb the balance arm. Prevent air currents from disrupting the balance arm.
Don’t set up the balance near one of the walls, the gravitational field from all that rock and brick will mask that of the test masses, and the balance will assume a "gravity gradient" position with one of the ends of the bar pointing toward the wall, and will budge only slightly under the influence of the test masses. With the bar in the middle of the room, the tidal influence of the mass of the wall and the rock behind it is reduced to a negligible value.
Use an aluminium ladder or a similar movable support frame to set up the balance in the middle of the room.
The Balance Arm and Cradle
The balance arm is a 5 × 5 × 30 cm bar of plastic foam, hacked from a 5 cm thick slab of packing material with a knife. The bar is suspended in a cradle made of insulated telephone wire. The bar is held in its cradle by friction and the indentation made in the soft plastic foam due to the weights at either end of the bar; it's easier to adjust the bar for proper alignment this way than if it were glued to the cradle.
The Support Fibre
The nylon monofilament that suspends the cradle is barely visible at the top of the picture - it is fastened by a knot to a loop formed into the cradle wires by twisting them. We use nylon monofilament because it closely approaches the ideal of a massless support free of torsional resistance. The masses which cause the bar to turn when a gravitational force acts upon them are lead "sinkers" used by fishermen. Two are placed on each end of the balance beam. Be sure to place the weights on both ends of the beam simultaneously so it doesn't topple, then adjust the placement so the beam is horizontal.
The Water Brake
The height of the beam is important because of the need for it to fit properly with the water brake (the tuna can seen at the bottom of the image above). If the beam is allowed to swing freely, it will be terribly underdamped - once it starts to swing, only air friction and the minuscule tension in the fibre will act to stop it. This causes the beam to bounce around incessantly, masking the steady influence of gravitation. The water brake dissipates the energy of these unwanted oscillations precisely as an automotive shock absorber does; the flap's motion does work on a viscous fluid, water in this case, and deposits its energy in heating it. The water brake consists of a flap which projects downward from the balance arm (in this case, a piece of aluminium cut with scissors from the tray of a "heat and eat" meal, fixed with white glue into a slot cut into the bottom of the balance beam). The flap projects into a reservoir (a tuna fish can) filled with water.
Test Masses and Supports
Ensure that the centre of gravity of the test masses are at the same height as the masses at the ends of the balance beam, maximising the attraction. Use the densest objects you can obtain for the ends of the balance beam and as test masses: lead sinkers, steel balls, plutonium hemispheres, etc. Density is important because the gravitational force varies as the inverse square of the distance between the centres of mass of two objects. With a dense substance, the centre of mass is closer to the surface, so you can get the centres of mass closer together and enhance the gravitational force. It's best to use a nonmagnetic material like lead for the weights on the ends of the balance arm. The forces we're working with are so small that if you use, for example, steel ball bearings on the arms, you may end up accidentally reinventing the compass instead of detecting the force of gravity.
Bending Spacetime in the Basement - John Walker
Tthe Michell-Cavendish Experiment - Laurent Hodges
Helping 'Big G' Get Back on Track - John Moore
Background to Boys' experiment to determine G - The Clarendon Laboratory Archive
Cavendish and G - Physical Sciences Phy Sci 119a
Weighing the Earth - The Book of Phyz, Dean Baird
The Weight of Expectation - C. D. Hoyle
Gravitational Torsion Balance: Instruction Manual - PASCO
Measurement of the Gravitational Constant - Rice University
The Controversy over Newton's Gravitational Constant - The UW Eot-Wash Group
Henry Cavendish and "G" - Chris Butlin
Cavendish Experiment - Harvard
